discrete valuation domain R is interesting, since every torsion-free module of ...... exercises, M and N are torsion-free modules of finite rank. ...... continuous if we assume that the group Î is equipped with the discrete topology. ...... A. G. Kurosh, âPrimitive torsionsfreie abelsche gruppen vom endlichen range,â Ann. Math., 38,.
Journal of Mathematical Sciences, Vol. 151, No. 5, 2008
MODULES OVER DISCRETE VALUATION DOMAINS. II UDC 512.553
P. A. Krylov and A. A. Tuganbaev
Abstract. In the second part of the paper, we study torsion-free modules and mixed modules. We analyze the possibility of the isomorphism of two modules with isomorphic endomorphism rings. We touch on several questions about transitive and fully transitive modules.
CONTENTS Chapter 5. Torsion-Free Modules . . . . . . . . . . . . . . . . . . . . . . . 23. Elementary Properties of Torsion-Free Modules . . . . . . . . . . . . . 24. The Category of Quasihomomorphisms . . . . . . . . . . . . . . . . . . 25. Purely Indecomposable and Copurely Indecomposable Modules . . . . 26. Indecomposable Modules over Nagata Valuation Domains . . . . . . . Chapter 6. Mixed Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 27. Uniqueness and Refinements of Decompositions in Additive Categories 28. Isotype, Nice, and Balanced Submodules . . . . . . . . . . . . . . . . . 29. Categories Walk and Warf . . . . . . . . . . . . . . . . . . . . . . . . . 30. Simply Presented Modules . . . . . . . . . . . . . . . . . . . . . . . . . 31. Decomposition Bases and Extension of Homomorphisms . . . . . . . . 32. Warfield Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 7. Definability of Modules by Their Endomorphism Rings . . . . 33. Theorems of Kaplansky and Wolfson . . . . . . . . . . . . . . . . . . . 34. Theorems of a Topological Isomorphism . . . . . . . . . . . . . . . . . 35. Modules over Completions . . . . . . . . . . . . . . . . . . . . . . . . . 36. Endomorphisms of Warfield Modules . . . . . . . . . . . . . . . . . . . Chapter 8. Modules with Many Endomorphisms or Automorphisms . . . . 37. Transitive and Fully Transitive Modules . . . . . . . . . . . . . . . . . 38. Transitivity over Torsion and Transitivity Mod Torsion . . . . . . . . 39. Equivalence of Transitivity and Full Transitivity . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The first part of this work was published earlier in the same journal; it contains Chapters 1–4 (Secs. 1–22); see J. Math. Sci., 145, No. 4, 4997–5117 (2007). The second part contains Chapters 5– 8 (Secs. 23–39).
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 122, Algebra, 2006. c 2008 Springer Science+Business Media, Inc. 1072–3374/08/1515–3255
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Chapter 5 TORSION-FREE MODULES In Chapter 5, we consider the following topics: elementary properties of torsion-free modules (Sec. 23); the category of quasihomomorphisms (Sec. 24); purely indecomposable and copurely indecomposable modules (Sec. 25); indecomposable modules over Nagata valuation domains (Sec. 26). Chapter 5 is devoted to torsion-free modules, which form one of the three main classes of modules. The other two classes are the class of primary modules and the class of mixed modules (see Sec. 4 (Part I)). Features of the studies of primary modules are mentioned in the Introduction and mixed modules are studied in the next chapter. Modules of finite rank are mainly considered. Only the case of an incomplete discrete valuation domain R is interesting, since every torsion-free module of finite rank over a complete domain is free (Corollary 11.7 (Part I)). There is an essential distinction between primary modules and torsion-free modules. As a rule, primary modules have a sufficiently large (in a certain sense) number of endomorphisms (see Corollary 7.5 and Corollary 7.6 in Part I and Proposition 37.5). On the contrary, torsion-free modules very often have few endomorphisms, and the endomorphism ring weakly affects the original module in the general case (for example, from the standpoint of the topics considered in Chapter 7). Any indecomposable primary module is a module of rank 1 (Corollary 11.8 (Part I)), whereas indecomposable torsion-free modules can have arbitrarily large finite ranks (see Example 11.9 (Part I) and remarks at the end of Sec. 26). As for direct decompositions, the existence of large indecomposable, torsion-free modules is the main difference between torsion-free modules and primary modules. However, direct decompositions of primary or torsionfree modules (even in the case of torsion-free modules of finite rank) can have various unexpected properties (see remarks to Chapters 4 and 5; see also Secs. 75, 90, and 91 of the work of Fuchs [90]). In Sec. 23, we present several standard properties of torsion-free modules. In Sec. 24, we consider the category of torsion-free modules; morphisms of this category are called quasihomomorphisms. This category differs from the ordinary category of modules considered in Secs. 13 and 22 (Part I), since we consider other morphisms. In the next chapter, we also consider two similar categories. In Sec. 25, we study modules of “smallest” or “largest” (in some sense) p-rank. In Sec. 26, indecomposable torsion-free modules of finite rank are considered. In addition, examples of indecomposable modules are also obtained. In this chapter and all remaining chapters, we assume that any discrete valuation domain R is commutative. The advantages of such an assumption were discussed in the beginning of Sec. 19 (Part I). K, and K. Precisely, R is the p-adic completion of the ring R; K and K We preserve the notation R, are the fields of fractions of the rings R and R, respectively. These symbols are used without additional explanations. 23.
Elementary Properties of Torsion-Free Modules
Here, we collect some results on torsion-free modules considered earlier in Secs. 4 and 9 (Part I) and present several new results. As earlier, denote by Hom(M, N ) and M ⊗ N the homomorphism group and the tensor product for the R-modules M and N , respectively. We canonically consider both of these groups as R-modules (see Secs. 2 and 13 (Part I)). In particular, the ring End(M ) is an R-algebra; this was already used in Chapter 4 (Part I). The K-space K ⊗ M and the Rp -space Rp ⊗ M will be very useful (Rp is the residue field of the ring R). We can identify the K-spaces K ⊗ M and KM , where KM is defined in Sec. 4 (Part I) (it is the set of fractions of a certain form). If M is a torsion-free module, then we also identify M with the image of the embedding M → KM , x → 1 · x, x ∈ M (Lemma 4.3 (Part I)). Then KM is a minimal K-space containing M and KM is the divisible hull of the module M (Proposition 6.8 (Part I)). Therefore, an 3256
arbitrary module M is a torsion-free module of finite rank iff M is isomorphic to a submodule of some finite-dimensional vector space over K. By Proposition 4.6 (Part I), the modules R and K are the only torsion-free modules of rank 1. Every proper submodule of the module K is isomorphic to R. If M and N are torsion-free modules, then it is obvious that Hom(M, N ) is also a torsion-free module. Consequently, we can take the divisible hull K Hom(M, N ) of it. The divisible module KN is injective by Theorem 6.5 (Part I). Therefore, every homomorphism from M into N can be extended to a unique K-homomorphism (linear operator) from KM into KN . The meaning of the inclusion Hom(M, N ) ⊆ HomK (KM, KN ) is clear. In this case, Hom(M, N ) = {f ∈ HomK (KM, KN ) | f M ⊆ N }. Since HomK (KM, KN ) is a K-space, we have that K Hom(M, N ) can be naturally embedded in HomK (KM, KN ). Precisely, any element (r/pn )ϕ ∈ K Hom(M, N ) (r/pn ∈ K, ϕ ∈ Hom(M, N )) corresponds to a homomorphism KM → KN such that ((r/pn )ϕ)((s/pm )x) = (rs/pn+m )ϕ(x) for all s/pm ∈ K and x ∈ M . Under such a correspondence, we have K Hom(M, N ) = {f ∈ HomK (KM, KN ) | pn f ∈ Hom(M, N ) for some n ≥ 0}. The rank r(M ) of the module M was defined in Sec. 4 (Part I). For the torsion-free module M , we have r(M ) = dimK (KM ) (Lemma 4.5 (Part I)). The p-rank rp (M ) of the module M was also introduced in Sec. 4 (Part I); it is equal to the dimension of the Rp -space M/pM (see also Property 6 in Sec. 9 (Part I) and the remark after it). If A is a pure submodule of the module M , then it is easy to see that r(M ) = r(A) + r(M/A) and rp (M ) = rp (A) + rp (M/A). We find the relation between the rank and the p-rank. First, it follows from Property 6 of Sec. 9 (Part I) that rp (M ) = r(B) for every basis submodule B of the module M . Therefore, the relation rp (M ) = 0 holds iff M is a divisible module, and rp (M ) = r(M ) iff M is a free module. Proposition 23.1. Let M be a torsion-free module of finite rank, and let F be some free submodule in M of maximal rank. Then M/F = C ⊕ D, where C is a direct sum of finitely many cyclic primary modules, D is a divisible primary module, and r(M ) = rp (M ) + r(D). Proof. First, we note that each such submodule F is generated by some maximal linearly independent system of elements of the module M . The converse assertion is also true (see Sec. 4 (Part I)). Consequently, r(F ) = r(M ) and M/F is a primary module. We choose some basis submodule B of the module M . A free basis of the module B (it is called a p-basis of the module M ) can be extended to a maximal linearly independent system of elements of the module M (see Secs. 5 and 9 in Part I). Therefore, it is clear that there exists a free module B0 such that B ⊕ B0 is a free submodule of maximal rank in M . We denote it by F0 ; we also consider the factor module M/F0 . It follows from the relations r(M ) = r(F ) and rp (M ) = r(B) that r(M ) − rp (M ) = r(B0 ). The factor module M/B is a divisible torsion-free module (i.e., a K-space) of rank r(M ) − rp (M ). We have M/F0 ∼ = (M/B)/(F0 /B), where F0 /B is a free module of rank r(M ) − rp (M ). Therefore, M/F0 is a divisible primary module of rank r(M ) − rp (M ). We pass to the factor module M/F . Let a1 , . . . , an and b1 , . . . , bn be free bases of the modules F0 and F , respectively, where n = r(M ). There exists an integer k ≥ 0 such that pk ai ∈ F and pk bi ∈ F0 for all i. Then we have pk F0 ⊆ F and pk F ⊆ F0 . Consequently, F0 /(F ∩ F0 ) and F/(F ∩ F0 ) are direct sums of finitely many cyclic primary modules. Considering the isomorphisms M/(F ∩ F0 ) ∼ = (M/F0 )/(F0 /(F ∩ F0 )) and
M/(F ∩ F0 ) ∼ = (M/F )/(F/(F ∩ F0 )), 3257
we obtain that the module M/(F ∩ F0 ) is isomorphic to the divisible module M/F0 . The module M/F is equal to C ⊕ D, where D ∼ = M/F0 and C is the module C indicated in the proposition. Let M be a torsion-free module of finite rank and let A be a submodule of M such that M/A is a primary module. Then r(M ) = r(A) and rp (M ) ≤ rp (A). The relation r(M ) = r(A) is obvious. We use Proposition 23.1 to verify the inequality. Lemma 23.2. For every module M , there exists an isomorphism Rp ⊗ M ∼ = M/pM . Therefore, rp (M ) = dimFp (Rp ⊗ M ). Proof. Let us consider the induced exact sequence of modules Rp ⊗ pM → Rp ⊗ M → Rp ⊗ M/pM → 0. Here, the first mapping is the zero mapping. Consequently, we obtain Rp ⊗ M ∼ = Rp ⊗ M/pM ∼ = M/pM. Proposition 23.3. If M and N are two torsion-free modules of finite rank, then (1) r(Hom(M, N )) ≤ r(M ) · r(N ) and rp (Hom(M, N )) ≤ rp (M ) · rp (N ); (2) M ⊗ N is a torsion-free module, r(M ⊗ N ) = r(M ) · r(N ),
and
rp (M ⊗ N ) = rp (M ) · rp (N ).
Proof. (1) We have r(Hom(M, N )) = dimK (K Hom(M, N )) ≤ dimK (Hom(KM, KN )) = dimK (KM ) · dimK (KN ) = r(M ) · r(N ). (2) There exists an embedding of Rp -spaces Rp ⊗ Hom(M, N ) → HomRp (Rp ⊗ M, Rp ⊗ N ). An arbitrary element of Rp ⊗ Hom(M, N ) is equal to 1 ⊗ ϕ. With any element 1 ⊗ ϕ, we associate the induced homomorphism Rp ⊗ M → Rp ⊗ N defined by the relation s ⊗ m → s ⊗ ϕ(m), s ∈ Rp , m ∈ M . In this way, we obtain an embedding. Then we use Lemma 23.2 and repeat the calculations used for Hom. (3) Taking induced exact sequences, we can embed M ⊗ N in the K-space KM ⊗ KN . We have r(M ⊗ N ) = dimK (K ⊗ M ⊗ N ) = dimK (K ⊗ M ) · dimK (K ⊗ N ) = r(M ) · r(N ). Replacing K by Rp , we obtain similar relations for p-ranks. For the modules M and N from the proposition, we obtain that Hom(M, N ) is a torsion-free module of finite rank and K Hom(M, N ) is a finite-dimensional K-space. In particular, K End(M ) is a finitedimensional K-algebra. Exercise 1. For every torsion-free R-module M of finite rank, there exists a pure submodule A of finite with rp (M ) = rp (Hom(M, A)). rank in R 24.
The Category of Quasihomomorphisms
The main result of the section is Theorem 24.9 on the existence of some duality for torsion-free modules of finite rank. In addition, category-theoretical notions related to this theorem are interpreted in terms of the modules. Additive categories and direct sums of objects were defined in Sec. 1 (Part I). We also define projections for direct sums. Lemma 24.1. An object A of the additive category E and morphisms ei ∈ HomE (Ai , A) (i = 1, . . . , n) form the direct sum of objects A1 , . . . , An with embeddings e1 , . . . , en iff for every i, there exists a morphism qi ∈ HomE (A, Ai ) such that ej qi = 0 for i = j, ei qi = 1Ai , and 1A = q1 e1 + · · · + qn en . In this case, qi ei is an idempotent of the ring EndE (A) for every i = 1, . . . , n. 3258
Proof. Necessity. For a fixed subscript i, we define dji ∈ HomE (Aj , Ai ) setting dii = 1Ai and dji = 0 if j = i, j = 1, . . . , n. By the definition of the direct sum, there exists a morphism qi ∈ HomE (A, Ai ) with ej qi = dji for every j. Then qi ei is an idempotent of the ring EndE (A) for every i = 1, . . . , n. It follows from the relations ei (q1 e1 + · · · + qn en ) = ei = ei 1A and the uniqueness property for direct sums that 1A = q1 e1 + · · · + qn en . Sufficiency. Assume that we have morphisms fi ∈ HomE (Ai , B) for every i = 1, . . . , n. We set f = q1 f1 + · · · + qn fn : A → B. Then ei f = fi for all i. Furthermore, if the morphism g ∈ HomE (A, B) has the property ei g = fi for every i, then 1A g = q1 (e1 g) + · · · + qn (en g) = q1 f1 + · · · + qn fn = f. Consequently, A is a direct sum of objects A1 , . . . , An with embeddings e1 , . . . , en . The morphisms q1 , . . . , qn are called the projections of the direct sum A = A1 ⊕· · ·⊕An with embeddings e1 , . . . , en . We say that idempotents split in the category E if for every object A and every idempotent e ∈ EndE (A), there exist an object B in E and morphisms f ∈ HomE (B, A) and g ∈ HomE (A, B) such that f g = 1B and gf = e. If idempotents split in the category E, then it is easy to verify that an object A of E is indecomposable iff 0 and 1A are the only idempotents of the ring EndE (A) (see Bass [31], Faith [68], and Arnold [6]). Let us formulate a known theorem on the uniqueness of decompositions for additive categories. The theorem deals with the local rings defined before Proposition 3.1 (Part I). In Sec. 27, two stronger results are proved. Remarks to this chapter and the end of Sec. 27 contain some information about the topics related to uniqueness and other properties of direct decompositions. Theorem 24.2 (Bass [31], Faith [68], and Arnold [6]). Let E be an additive category, and let idempotents split in E. We assume that the direct sum A = A1 ⊕ · · · ⊕ An is given in E, where every EndE (Ai ) is a local ring. Then (1) if A = B1 ⊕ · · · ⊕ Bm , then every object Bj is a finite direct sum of indecomposable objects in E; (2) in (1), if every Bj is indecomposable, then m = n, and there exists a permutation θ of the set {1, 2, . . . , n} such that Ai is isomorphic to Bθ(i) for every i. We return to torsion-free modules. Let M and N be two torsion-free modules. The elements of the K-space K Hom(M, N ) are called quasi-homomorphisms (see the previous section in connection with the notation K Hom(M, N )) and the space K Hom(M, N ) is called the quasi-homomorphism group. We define the category T F as follows. Torsion-free modules of finite rank are objects of T F and quasihomomorphisms are morphisms of T F. Proposition 24.3. The category T F is additive and idempotents are split in T F. Proof. Define the composition K Hom(M, N ) × K Hom(N, L) → K Hom(M, L) by the relation (k1 ϕ, k2 ψ) → k1 k2 (ϕψ). This composition is associative, and 1 · 1M is the identity element, which is also denoted by 1M . We know that K Hom(M, N ) is an Abelian group, and it is easy to verify that the composition of quasi-homomorphisms is bilinear. There exist finite direct sums in T F. Let A1 , . . . , An be objects in T F. Take the module direct sum M = A1 ⊕ · · · ⊕ An and natural embeddings κi : Ai → M , i = 1, . . . , n. We verify that M is a direct sum of the objects A1 , . . . , An with embeddings κ1 , . . . , κn in T F. Assume that have quasihomomorphisms fi ∈ K Hom(Ai , N ) for some torsion-free module N , where i = 1, . . . , n. We choose a nonnegative integer n with pn fi ∈ Hom(Ai , N ). There exists a unique homomorphism ϕ : M → N such that κi ϕ = pn fi for 3259
all i. Then (1/pn )ϕ ∈ K Hom(M, N ) and κi ((1/pn )ϕ) = fi for all i. The uniqueness of (1/pn )ϕ follows from the uniqueness of ϕ. Thus, T F is an additive category. Let ε = (r/pn )α ∈ K End(M ) for some r/pn ∈ K, and let α ∈ End(M ) be an idempotent of the ring K End(M ). We need to prove the existence of a torsion-free module A of finite rank and quasihomomorphisms κ ∈ K Hom(A, M ), π ∈ K Hom(M, A) such that πκ = ε and κπ = 1A . Let A = αM , let i : A → M be the natural embedding, κ = (1/pn )i, and let π = rα. Then πκ = (r/pn )(αi) = ε. It follows from the relation ε2 = ε that r2 α2 = rpn α. Now, let a ∈ A and a = αx, where x ∈ M . We have (rpn )(κπ)a = r2 α2 (x) = rpn α(x) = rpn a. Therefore κπ = 1A and the idempotent ε splits. The category T F is called the category of quasi-homomorphisms (of torsion-free modules of finite rank). The following terminology is used in the literature. The isomorphisms of the category T F are called quasi-isomorphisms. If two modules M and N are isomorphic to each other in T F, then we say that M and N are quasi-isomorphic to one another and write M ∼ N . If M = A1 ⊕· · ·⊕An in T F, then M is a quasi-direct sum of the modules Ai which are called quasi-summands. Modules indecomposable in T F are called strongly indecomposable modules. The projections from Lemma 24.1 are called quasi-projections. The endomorphism ring K End(M ) of a module M in T F is called the quasi-endomorphism ring of M . For the subsequent presentation, it is important that the K-algebra K End(M ) be finite-dimensional (see the end of Sec. 23). We formulate the main notion and properties of the category T F in purely module-theoretical terms. Corollary 24.4. For any two torsion-free modules M and N of finite rank, the following conditions are equivalent: (1) M is quasi-isomorphic to N ; (2) there exist homomorphisms ϕ : M → N and ψ : N → M and an integer n ≥ 0 such that ϕψ = pn 1M and ψϕ = pn 1N ; (3) there exist monomorphisms ϕ : M → N and ψ : N → M such that the modules N/ Im ϕ and M/ Im ψ are bounded ; (4) there exist a submodule N and an integer n such that pn N ⊆ N ⊆ N and M ∼ = N ; m (5) there exist submodules M and N and integers m and n such that p M ⊆ M ⊆ M , pn N ⊆ N ⊆ N , and M ∼ = N . Proof. (1) =⇒ (2). The modules M and N are isomorphic to one another in T F. There exist f ∈ K Hom(M, N ) and g ∈ K Hom(N, M ) with f g = 1M and gf = 1N in T F. We choose an integer m ≥ 0 such that pm f = ϕ ∈ Hom(M, N ) and pm g = ψ ∈ Hom(N, M ). Then ϕψ = p2m 1M and ψϕ = p2m 1N . (2) =⇒ (3). It is clear that ϕ and ψ are monomorphisms and pn N = Im(ψϕ) ⊆ Im(ϕ). Similarly, n p M ⊆ Im(ψ). (3) =⇒ (4). We take an integer n with pn N ⊆ Im(ϕ) and set N = Im(ϕ). The implication (4) =⇒ (5) is obvious. (5) =⇒ (1). Let ϕ : M → N and ψ : N → M be mutually inverse isomorphisms. Then (1/pm )(pm ϕ) and (1/pn )(pn ψ) are morphisms from K Hom(M, N ) and K Hom(N, M ), respectively; the compositions of these morphisms in T F are equal to 1M and 1N , respectively. It is obvious that two isomorphic modules are quasi-isomorphic to one another. If the both modules M and N are free or divisible and they are quasi-isomorphic to one another, then M and N are isomorphic to one another. This is not true for arbitrary modules (see Example 25.7). The rank and the p-rank of the module are preserved under quasi-isomorphisms. As for the rank, this follows from property (4); as for the p-rank, this follows from Proposition 23.1. 3260
Corollary 24.5. For a torsion-free module M of finite rank, the following assertions (a), (b) and (c) hold. (a) Let M = A1 ⊕ · · · ⊕ An be a direct sum in T F, let ei and qi be the morphisms from Lemma 24.1, and let qi ei = (ri /pki )εi , where ri /pki ∈ K, εi ∈ End(M ), i = 1, . . . , n. There exists an integer m ≥ 0 such that pm M ⊆ ε1 (M ) ⊕ · · · ⊕ εn (M ). Furthermore, if κi : εi (M ) → M is a natural embedding, then M is the direct sum in T F of the modules ε1 (M ), . . . , εn (M ) with embeddings κ1 , . . . , κn . (b) The following conditions are equivalent: (1) M is a strongly indecomposable module; (2) A = 0 or B = 0 in each case, where pm M ⊆ A ⊕ B ⊆ M ; (3) K End(M ) is a local ring. (c) There exist an integer m ≥ 0 and strongly indecomposable modules A1 , . . . , An such that pm M ⊆ A1 ⊕ · · · ⊕ An ⊆ M. Proof. (a) By Lemma 24.1, we have 1M = q1 e1 +· · ·+qn en , where qi ei is a pairwise orthogonal idempotent of the ring K End(M ). It is easy to see that the sum ε1 (M ) + · · · + εn (M ) is a direct sum. Let m be the maximum of the integers k1 , . . . , kn . Then for every x ∈ M , we have pm x = pm (q1 e1 + · · · + qn en )(x) ∈ ε1 (M ) ⊕ · · · ⊕ εn (M ). The second assertion is directly verified. Therefore, the initial direct sum M = A1 ⊕ · · · ⊕ An can be replaced in T F by the direct sum M = ε1 (M ) ⊕ · · · ⊕ εn (M ). (b) The equivalence of (1) and (2) follows from (a). If (1) holds, then M is indecomposable in T F. Consequently, 0 and 1M are the only idempotents of the ring K End(M ). This implies that the finitedimensional K-algebra K End(M ) is local. Conversely, the local ring K End(M ) has no nontrivial idempotents and M is indecomposable in T F. The proof of assertion (c) uses (b) and the induction on the rank of the module M . It follows from assertions 24.2–24.5 that the category T F satisfies the isomorphism theorem for direct decompositions with indecomposable summands. We formulate this theorem without category T F. Corollary 24.6. Let M be a torsion-free module of finite rank (k ≥ 0), and let A1 , . . . , An be strongly indecomposable modules such that p k M ⊆ A 1 ⊕ · · · ⊕ An ⊆ M (see Corollary 24.5 (c)). (1) If l ≥ 0 and pl M ⊆ B1 ⊕ · · · ⊕ Bm ⊆ M, then every module Bj is quasi-isomorphic to the direct sum of some modules Ai ; (2) if each of the modules Bj is strongly indecomposable, then m = n, and there exists a permutation θ of the set {1, . . . , n} such that Ai is quasiisomorphic to Bθ(i) for every i. Any strongly indecomposable module is indecomposable. In general, the converse is not true. Before we consider the duality mentioned in the beginning of the section, we do some preliminary work. of the domain R and the field of fractions K of the domain R. We We will use the p-adic completion R ⊂K and R ⊂ K ⊆ K, where R ∩ K = R. Furthermore, keep in mind the known ring inclusions R ⊆ R R/R is a divisible torsion-free R-module (see the paragraph before Corollary 11.15 (Part I)). Every Rtorsion-free module of finite rank is free. If X is a subset of a module A over some ring S, then SX denotes the S-submodule in A generated by X. Assuming a homomorphism of S-modules, we say an “S-homomorphism.” In what follows, we consider some finite-dimensional vector K-spaces. Let V be a finite-dimensional vector K-space, X be some basis of V , V = KX, and let X = X1 ∪ X2 , 2 → RX 1 is an R-homomorphism. where X1 ∩ X2 = ∅. Then V = KX1 ⊕ KX2 . We assume that α : RX 3261
i is the divisible hull of RX i , α extends to a unique K-homomorphism Since KX (a linear operator) KX2 → 1 , which is also denoted by α. With respect to the above decomposition, the matrix 1 0 defines KX α 1 1 ⊕ KX 2 ). the automorphism α of the K-space V . By (X1 , X2 , V , α ), we denote the R-module α (V ) ∩ (RX ). Then M is a torsion-free module of finite rank, α (RX) is a Lemma 24.7. Let M = (X1 , X2 , V , α free submodule in M of maximal rank, KM = α (V ), RX1 is a basis submodule of the module M , and = RX 1 ⊕ KX 2 . If M is a reduced module, then α(y) = 0 for all nonzero y ∈ KX2 . RM Proof. We have that V and α (V ) are K-spaces of dimension |X|. Consequently, M is a torsion-free R-module of finite rank. Since RX is a free module, α (RX) also is a free module of rank |X| and α (RX) ⊆ M . It follows from the inclusion M ⊆ α (V ) that KM ⊆ α (V ). Consequently, these K-spaces coincide, since they are of equal dimension. Since α (RX1 ) = RX1 , RX1 ⊆ M and RX1 is a free module 1 ⊕ KX 2 and (RX 1 ⊕ KX 2 )/RX1 is a divisible of rank |X1 |. Furthermore, M is a pure submodule in RX 2 (see Exercise 5 (b)). Therefore, M/RX1 1 /RX1 ⊕ KX torsion-free module, which is isomorphic to RX is a divisible torsion-free module and RX1 is a basis submodule of the module M . The purity of M in 2 implies the purity of RM in KX 1 ⊕ KX 2 (Exercise 6). Consequently, RM coincides with 1 ⊕ KX RX 1 ⊕ KX 2 , since they are of equal rank as R-modules. RX Let M be a reduced module, y ∈ KX2 , and α(y) = 0. Then α (y) = (1 + α)(y) = y. Consequently, Ky ⊂ M and y = 0, since Ky ∼ = K for y = 0. Every torsion-free module of finite rank can be constructed by using the construction considered before Lemma 24.7. Lemma 24.8. Every torsion-free module M of finite rank is equal to (X1 , X2 , V , α ) for some X1 , X2 , V , and α. Proof. Choose some p-basis X1 of the module M . Then RX1 is a basis submodule of the module M . We enlarge X1 up to a maximal linearly independent set X1 ∪ Y of the module M . Define the vector K-space 1 ⊕ KY with basis X1 ∪ Y . Then KM = KX1 ⊕ KY . Since RX1 is a basis submodule of the V = KX (Exercise 5 (c)). We have a decomposition module M , RX1 is a basis submodule of the R-module RM = RX 1 ⊕ D, where D is the maximal divisible submodule of the R-module (note that RX 1 is a RM RM complete module). 2 → RX 1 such that α : RX Let us find a linearly independent subset X2 in V and an R-homomorphism = RX 1 ⊕ KX 2 and (1 + α)(X2 ) = Y. RM on We denote by π1 and π2 the projections from RM to RY is injective. We set X2 = π2 (Y ). Then X2 is independent set and |Y | = dim D. Consequently, 2 = D and RM KX
1 and D, respectively. The restriction of π2 RX a basis of the K-space D, since Y is a linearly
1 ⊕ KX 2. = RX 2 → RY and π1 : RY → RX 1 . If y ∈ Y and Let α be the composition of the mappings π2−1 : RX 1 , x ∈ D, then π2 (y) = x ∈ X2 . Then (1 + α)x = y, and it is easy to verify that y = z + x, where z ∈ RX (1 + α)(X2 ) = Y . ), which is denoted by N . We have Finally, let V = KX1 ⊕ KX2 . Consider the module (X1 , X2 , V , α 2 ), 1 ⊕ KX N =α (V ) ∩ (RX 1 0 . It follows from (1 + α)(X2 ) = Y that where α = α 1 α (V ) = KX1 ⊕ KY = KM 3262
and M ⊆ α (V ).
Thus, M ⊆ N . However, N/M is a primary module, since (KX1 ⊕KY )/M is primary. On the other hand, N/M is a torsion-free module, since the module RM/M is torsion-free (Exercise 5 (b)). Consequently, N = M. We take two modules
, β); ) and N = (Y1 , Y2 , U M = (X1 , X2 , V , α . Let investigate the interrelations between quasihomomorphisms M → N and K-homomorphisms V → U ), there exists a unique f : M → N be some quasihomomorphism. Since KM = α (V ) and KN = β(U ) that extends f . We have that α 1 ∪Y2 ) are homomorphism of K-spaces f : α (V ) → β(U (X1 ∪X2 ) and β(Y bases of K-spaces V and U , respectively. Therefore, there exists a unique K-homomorphism f : V → U which extends f . Consequently, f extends f . We choose a nonzero element r ∈ R with (rf )M ⊆ N . ) ⊆ RN . Therefore, f(KX 2 ) ⊆ KY 2 (KX 2 and KY 2 are maximal divisible submodules of Then (rf )(RM and RN , respectively). modules RM ) and g(KX be a K-homomorphism 2 ) ⊆ KY 2 . For Conversely, let g : V → U such that g( α(V )) ⊆ β(U R-modules, we have = RX 1 ⊕ KX 2, = RY 1 ⊕ KY 2. RM RN
) = KN . It is clear that there exists a nonzero element r ∈ R such that Recall that α (V ) = KM and β(U and (rg)(RM ) ⊆ RN . Therefore, (rg)M ⊆ N (see Exercise 7). We set f = g|M . Then (rg)(X1 ) ⊆ RN f : M → N is a quasihomomorphism. Note that a quasihomomorphism f : M → N is a quasiisomorphism iff both mappings f : α (V ) → β(U ) and f : KX2 → KY2 are isomorphisms. Finally, f is an isomorphism iff f : α (V ) → β(U ) and → RN are isomorphisms. f : RM We have completed the preliminary work and return to the category T F. Below, we use the notions of a contravariant functor, a natural equivalence, and other notions; they are presented in Sec. 1 (Part I). A contrainvariant functor F : E → E is called a duality if F 2 is naturally equivalent to the identity functor of the category E. Recall the known duality for finite-dimensional vector spaces. Let K-mod be the category of finite dimensional vector K-spaces. For two finite-dimensional K-spaces V and U and a K-homomorphism ∗ ∗ ∗ and G(f ) = f , where f is the induced homomorphism f : V → U , we set G(V ) = V = HomK (V, K) ∗ ∗ ∗ ∗ U → V , f (ϕ) = f ϕ (ϕ ∈ U ; see Sec. 2 (Part I)). We obtain the duality G of the category K-mod. 2 → G is defined as follows. For the K-space V , the The corresponding natural equivalence h : 1K-mod natural isomorphism hV : V → V ∗∗ is defined by the relation (hV (a))(ψ) = ψ(a) for all a ∈ V and ψ ∈ V ∗ . The K-space V ∗ is said to be dual to V and the homomorphism f ∗ is said to be dual to f . Let X = (x1 , x2 , . . . , xn ) be some basis of the K-space V . For every i = 1, . . . , n, we denote by x∗i the element of the K-space V ∗ such that x∗i (xi ) = 1 and x∗i (xj ) = 0 for i = j (x∗i is a homomorphism We denote (x∗ , . . . , x∗ , x∗ ) by X ∗ . Here, X ∗ is a basis of the space V ∗ . V → K). n 2 1 Theorem 24.9 (Arnold [3]). There exists a duality F of the category T F such that for every torsion-free module M of finite rank, the following properties hold : (1) r(F (M )) = r(M ); (2) rp (F (M )) = r(M ) − rp (M ); (3) M is a free module iff F (M ) is a divisible module. Proof. Let us define the required contravariant functor F . Let M be a torsion-free module of finite rank and let M = (X1 , X2 , V , α ) (cf. Lemma 24.8). We set F (M ) = (X2∗ , X1∗ , (V )∗ , α ¯ ), where X2∗ ∪ X1∗ is ∗ the basis of the dual K-space (V ) defined in the way indicatedbefore the ¯ theorem. The mapping α ∗ 1 −α ∗ → RX ∗ is , where α∗ : RX coincides with (( α)∗ )−1 : (V )∗ → (V )∗ . The matrix form of α ¯ is 1 2 0 1 3263
2 → RX 1 . Note that KX ∗ = KX ∗ ⊕ KX ∗ ; KX ∗ can be identified the mapping which is dual to α : RX 2 1 ∗ with V = HomK (V, K), since X = X1 ∪ X2 is a basis of the K-space V . By Lemma 24.7, F (M ) is a torsion-free module of finite rank. of finite , β) Now assume that we have two torsion-free modules M = (X1 , X2 , V , α ) and N = (Y1 , Y2 , U rank and a quasihomomorphism f : M → N . We can assume that f is a K-homomorphism V → U . Let f ∗ : (U )∗ → (V )∗ be the dual homomorphism of K-spaces. and a K-homomorphism V → U We ∗ show that the restriction of the homomorphism f to F (N ) is a quasihomomorphism F (N ) → F (M ). For this purpose, we use the remarks after the proof of Lemma 24.8. Since f (V ) ⊆ U , f ∗ (U ∗ ) ⊆ V ∗ . We ∗ ) ⊆ KX ∗ . It suffices to prove that f ∗ (Y ∗ ) ⊆ KX ∗ . Let X1 = (x1 , . . . , xs ), need to verify that f ∗ (KY 1 1 1 1 X2 = (xs+1 , . . . , xm ), Y1 = (y1 , . . . , yt ), and Y2 = (yt+1 , . . . , yn ). We take some yj∗ ∈ Y1∗ , j = 1, . . . , t. We Let xi ∈ X2 and s + 1 ≤ i ≤ m. Then have f ∗ (yj∗ ) = q1 x∗1 + · · · + qm x∗m , where q1 , . . . , qm ∈ K. (f ∗ (yj∗ ))(xi ) = (f yj∗ )(xi ) = yj∗ (f (xi )) = 0, 2 and y ∗ annihilates KY 2 . On the other hand, since f ∗ (X2 ) ⊆ KY j (f ∗ (yj∗ ))(xi ) = q1 x∗1 (xi ) + · · · + qm x∗m (xi ) = qi . ∗. Consequently, qi = 0 for all i = s + 1, . . . , m and f ∗ (yj∗ ) ∈ KX 1 ∗ Now we set F (f ) = f : F (N ) → F (M ). The property that F is a contravariant functor in the category T F follows from the property that F is induced by the contravariant functor G on the category K-mod. 2 We prove that the functor F is naturally equivalent to the identity functor of the category T F. For ¯ ). We note that the the module M = (X1 , X2 , V , α ), the module F 2 (M ) has the form (X1∗∗ , X2∗∗ , (V )∗∗ , α ∗∗ ¯ is similar to the definition of α ¯ = ( definition of α ¯ and α α) . We take the natural K-homomorphism 2 ) = KX ∗∗ . Furthermore, we have hV : V → (V )∗∗ . Then hV (KX 2 hV ( α(V )) = hV ( α(KX1 ⊕ KX2 )) = ( α)∗∗ (hV (KX1 ⊕ KX2 ) ∗∗ ∗∗ ¯ = ( α)∗∗ (KX2∗∗ ⊕ KX1∗∗ ) = α(KX 2 ⊕ KX1 ). Let gM be the restriction of hV to M . By remarks after Lemma 24.8, we have that gM : M → F 2 (M ) is a quasiisomorphism, i.e., it is an isomorphism in T F. be some quasihomomorphism. As above, we consider it as a K , β) Let f : M → N = (Y1 , Y2 , U . The functor G2 and the identity functor of the category K-mod are naturally homomorphism V → U 2 2 equivalent. Therefore, f hU = hV G (f ). Therefore f gN = gM F (f ). Consequently, F 2 is naturally equivalent to the identity functor of the category T F. Properties (1) and (2) follow from Lemma 24.7 and the definition of the functor F . Now (3) is obvious. The duality F from Theorem 24.9 is called the Arnold duality. A short construction of the Arnold duality is given in the paper of Lady [186]. Exercise 1. Construct the indecomposable torsion-free module of finite rank that has a nontrivial quasidecomposition. is a completion of M in the p-adic topology. Exercise 2. If M is a reduced module, then RM Exercise 3 (Arnold [3]). Construct an example of isomorphic torsion-free modules M and N of rank 3 and p-rank 1 such that the modules F (M ) and F (N ) are not isomorphic to one another. (Since F is a duality, the modules F (M ) and F (N ) are quasiisomorphic to one another.) Exercises 4–7 are related to some details of the proofs of Lemma 24.7 and Lemma 24.8. In these exercises, M and N are torsion-free modules of finite rank. ⊗ M and RM are canonically isomorphic to one anther. Exercise 4. The R-modules R 3264
Exercise 5. Prove that is equal to the rank of the R-module M ; (a) the rank of the R-module RM (b) RM/M is a divisible torsion-free module; ; (c) every p-basis of the module M is a p-basis for the module RM ∩ KM = M . (d) RM is a pure submodule of the module Exercise 6. If A is a pure submodule in the module M , then RA RM . be a homomorphism of K-spaces Exercise 7. Let g : V → U considered in remarks after Lemma 24.8. If g(KM ) ⊆ KN and (rg)(RM ) ⊆ RN for some nonzero element r ∈ R, then (rg)M ⊆ N . Exercise 8. Prove that two torsion-free modules M and N of finite rank are quasiisomorphic to each other iff r(M ) = r(N ), rp (M ) = rp (N ), and M is isomorphic to some submodule in N . 25.
Purely Indecomposable and Copurely Indecomposable Modules
Under the Arnold duality, divisible modules correspond to free modules; this follows from Theorem 24.9. In this section, we consider and study two wider classes of modules that are dual to each other. For a torsion-free module M of finite rank, we always have the inequality rp (M ) ≤ r(M ). For rp (M ) = 0, we obtain divisible modules. For rp (M ) = r(M ), we obtain free modules. The objects of our attention are modules M such that rp (M ) = 1 or rp (M ) = r(M ) − 1. They have various remarkable properties. denotes the completion of a reduced torsion-free module M in the p-adic topology. As earlier, M If M and N are two such modules, then every homomorphism from M into N extends to a unique in N (Proposition 11.12 (Part I)). In addition, M and N are isomorphic to one homomorphism from M →N with ϕ(KM ) ⊆ KN (we assume that M ∩KM = M ; another iff there exists an R-isomorphism ϕ:M see also Exercises 5 and 7 in Sec. 24). A reduced torsion-free module M of finite rank is said to be purely indecomposable if every pure submodule of the module M is indecomposable. There are many characterizations of purely indecomposable modules. Proposition 25.1. For a reduced torsion-free module M of finite rank, the following conditions are equivalent: (1) (2) (3) (4) (5)
M is a purely indecomposable module; M is isomorphic to some pure submodule of the module R; rp (M ) = 1; if A is a proper pure submodule of the module M , then M/A is a divisible module; for any reduced torsion-free module N , every nonzero homomorphism M → N is a monomorphism.
Proof. (1) =⇒ (2). Every basis submodule B of the module M is isomorphic to R. Consequently, ∼ ∼ which implies (2) (see Sec. 11 (Part I)). M =B = R, = 1, rp (M ) = 1. (2) =⇒ (3). Since 0 < rp (M ) ≤ rp (R) (3) =⇒ (4). Since 1 = rp (M ) = rp (A) + rp (M/A) and rp (A) = 0, rp (M/A) = 0 and M/A is a divisible module. (4) =⇒ (5). We assume that there exists a nonzero homomorphism ϕ : M → N with Ker(ϕ) = 0. Then M/ Ker(ϕ) and Im(ϕ) are divisible modules, which is impossible. (5) =⇒ (1). We assume that M is not a purely indecomposable module. Then there exists a pure submodule A of the module M that has a nontrivial direct decomposition A = X ⊕ Y . The projection (Theorem 11.4 (Part I)) with nonzero from the module A onto X extends to a homomorphism M → M kernel, which contradicts (5). Consequently, M is a purely indecomposable module. 3265
If M is a purely indecomposable module, then it follows from rp (M ) = 1 that M/pk M ∼ = R(pk ) for all k ≥ 1. The irreduced torsion-free module M of finite rank with rp (M ) = 1 is equal to N ⊕ D, where N is a purely indecomposable module and D is a divisible module. Proposition 25.2. If M and N are purely indecomposable modules, then the following conditions are equivalent: (1) M ∼ = N; (2) M ∼ N ; (3) Hom(M, N ) = 0 and Hom(N, M ) = 0; (4) r(M ) = r(N ) and Hom(M, N ) = 0. Proof. The implication (1) =⇒ (2) is obvious. The implication (2) =⇒ (3) follows from Corollary 24.4. (3) =⇒ (4). It follows from Proposition 25.1 that there exist monomorphisms M → N and N → M . Consequently, r(M ) ≤ r(N ) ≤ r(M ) and r(M ) = r(N ). (4) =⇒ (1). As earlier, there exists a monomorphism from M into N . We assume that M is a submodule in N . Let G be a free submodule in M of maximal rank. It follows from Proposition 23.1 that M/G = C1 ⊕ D1 , where C1 is the direct sum of finitely many cyclic primary modules and D1 is a divisible primary module. Since r(M ) = r(N ), we have that N/G = C2 ⊕ D2 , where the structures of the summands C2 and D2 are similar to the structures of C1 and D1 . Furthermore, r(M ) = rp (M ) + r(D1 ) and r(N ) = rp (N ) + r(D2 ). Since rp (M ) = rp (N ), r(D1 ) = r(D2 ), i.e., D1 ∼ = D2 . It follows from the relation N/M ∼ = (N/G)/(M/G) that N/M is the direct sum of a finite number of cyclic primary modules. Therefore, there exists an integer k ≥ 0 with pk N ⊆ M . Since N/pk N is a cyclic module R(pk ), M/pk N = pm N/pk N , where m ≤ k and M = pm N . However, N ∼ = pm N under the correspondence x → pm x. Consequently, M ∼ = N. We can obtain sufficiently complete information on the endomorphism ring of a purely indecomposable module. As earlier (see Sec. 19 in Part I), we assume that the domain R is contained in the endomorphism ring of this torsion-free module (the element r ∈ R is identified with the endomorphism x → rx (x ∈ M ) of the module M ). If we use module terms for some R-algebra, we consider the algebra as an R-module. The symbol [P : K] denotes the degree of the extension of the field P over the field K. Proposition 25.3. Let M be a purely indecomposable module and let S = End(M ). Then the R-algebra and S is a discrete valuation domain with prime element p. S is isomorphic to some pure subalgebra in R [P : K] is finite, P = KS, and The field of fractions P of the domain S satisfies K ⊆ P ⊆ K, EndR (S) = EndS (S). It is clear that M is a dense Proof. By Proposition 25.1, we can assume that M is a pure submodule in R. submodule. Now we have the following property. An arbitrary endomorphism α ∈ S can be extended and the extension is unique (Proposition 11.12 (Part I)). In to an endomorphism α of the module R addition, α is an R-module endomorphism (Example 12.4 (Part I)). Consequently, α coincides with the on the ring R. Conversely, each element r ∈ R with rM ⊆ M yields multiplication by some element r ∈ R | rM ⊆ M }. In an endomorphism of the module M . The algebra S can be identified with the set {r ∈ R since M is pure in R. Let r be a nonzero element of S. We have addition, S is a pure subalgebra in R, n r = p v, where n ≥ 0 and v is an invertible element in R. Indeed, v ∈ S, since S is a pure subalgebra. Furthermore, vM is a pure submodule in M which is isomorphic to M . Consequently, vM = M , since the rank M is finite. Thus, r = pn v, where v is an invertible element in S. Now it is clear that the nonzero ideals of the ring S have the form pn S, n ≥ 0. Consequently, S is a discrete valuation domain with prime K ⊆ P ⊆ K. It follows from the relation P = {r(1/pn ) | r ∈ S, n ≥ 0} element p. Since R ⊆ S ⊆ R, 3266
that P = KS. The dimension of P over K is equal to the rank of the R-module S; this rank is finite (Proposition 23.3). Let α be some endomorphism of the R-module S. As above, α coincides with the on S. It follows from α(1) = r · 1 = r that r ∈ S. Consequently, α multiplication by some element r ∈ R is an S-endomorphism of the module S. Purely indecomposable modules are easily described up to isomorphism. We fix an integer n ≥ 1. Let We denote by A(a1 , . . . , an ) the 1, a1 , . . . , an be some linearly independent elements of the R-module R. generated by these elements. It is easy to verify that pure submodule in R A(a1 , . . . , an ) = A(r1 a1 , . . . , rn an ) for all nonzero elements ri ∈ R and A(a1 , . . . , an ) = KA(a1 , . . . , an ) ∩ R, where KA(a1 , . . . , an ) = K ⊕ Ka1 ⊕ · · · ⊕ Kan (see Exercise 5 in Sec. 24). is the K-subspace generated by A(a1 , . . . , an ) in K Let M be a purely indecomposable module of rank n + 1. Then M is isomorphic to the module Indeed, it follows from ProposiA(a1 , . . . , an ) for some linearly independent elements 1, a1 , . . . , an in R. We choose some nonzero element in M . tion 25.1 that we can assume that M is a pure submodule in R. k Since M is pure, we also This element is equal to p v, where k ≥ 0 and v is some invertible element in R. −1 −1 In addition, 1 = v −1 v ∈ v −1 M . have v ∈ M . We have M ∼ = v M , where v M is a pure submodule in R. Now it is clear that the module v −1 M coincides with A(a1 , . . . , an ) for some elements a1 , . . . , an . It follows from the proof of Proposition 25.3 that | rA(a1 , . . . , an ) ⊆ A(a1 , . . . , an )}. End(A(a1 , . . . , an )) = {r ∈ R If r ∈ End(A(a1 , . . . , an )), then r ∈ A(a1 , . . . , an ), since A(a1 , . . . , an ) contains the identity element. Therefore, we also have K End(A(a1 , . . . , an )) = {r ∈ K ⊕ Ka1 ⊕ · · · ⊕ Kan | rai ∈ K ⊕ Ka1 ⊕ · · · ⊕ Kan , 1 ≤ i ≤ n}. Theorem 25.4 (Arnold [3], Arnold–Dugas [12]). Two purely indecomposable modules A(a1 , . . . , an ) and A(b1 , . . . , bn ) are isomorphic to each other iff K has elements si and tji (0 ≤ i ≤ n, 1 ≤ j ≤ n) such that and ubj = tj0 + tj1 a1 + · · · + tjn an . u = s0 + s1 a1 + · · · + sn an is an invertible element in R Proof. Let ϕ : A(b1 , . . . , bn ) → A(a1 , . . . , an ) be some isomorphism and let u = ϕ(1). Similarly to the proof of Proposition 25.3, we can obtain that ϕ coincides with the multiplication by ϕ(1), i.e., ϕ(y) = u · y for all y ∈ A(b1 , . . . , bn ). It is clear that u is an invertible element and u, ubj ∈ A(a1 , . . . , an ); this implies the existence of the elements si and tji . Conversely, we assume that there exist indicated elements si and tji . We consider the multiplication by the element u. We choose pk such that pk u, pk (ubj ) ∈ A(a1 , . . . , an ) for all j. Then of the ring R u, ubj ∈ A(a1 , . . . , an ), since the submodule A(a1 , . . . , an ) is pure. Considering the purity, we obtain uA(b1 , . . . , bn ) ⊆ A(a1 , . . . , an ). Since uA(b1 , . . . , bn ) is a pure submodule in A(a1 , . . . , an ), the above inclusion is an equality (see also Proposition 25.2). Consequently, the multiplication by u is an isomorphism between A(b1 , . . . , bn ) and A(a1 , . . . , an ). A torsion-free module M is said to be coreduced if M has no direct summands that are free modules. The module M is coreduced iff Hom(M, R) = 0. Under the Arnold duality, reduced modules correspond to coreduced modules. If M is an arbitrary torsion-free module of finite rank, then M = N ⊕ G, where N is a coreduced module and either G is a free module or G = 0. A coreduced module M of finite rank is said to be copurely indecomposable if the factor module M/A is indecomposable for every pure submodule A. 3267
Characterizations of copurely indecomposable modules are dual to the characterizations of purely indecomposable modules from Proposition 25.1. Proposition 25.5. For a coreduced torsion-free module M of finite rank, the following properties are equivalent: (1) M is a copurely indecomposable module; (2) r(M ) = rp (M ) + 1; (3) every submodule A of the module M with r(A) < r(M ) is free; (4) F (M ) is a purely indecomposable module, where F is the Arnold duality from Theorem 24.9; (5) for each coreduced torsion-free module N and every nonzero homomorphism ϕ : N → M , the factor module M/ Im(ϕ) is a bounded module. Proof. (1) =⇒ (2). We assume that r(M ) = rp (M ) + k and k ≥ 2. We take some basis submodule B of the module M . Then we have r(M/B) = r(M ) − r(B) = rp (M ) + k − rp (M ) = k. Therefore, the divisible module M/B of rank k is a decomposable module, which contradicts (1). Consequently, k = 1. (2) =⇒ (3). Let A ⊆ M and r(A) < r(M ). We assume that A is pure in M ; otherwise, we can take the pure submodule generated by A in M (see property (7) from Sec. 7 (Part I)). We have the relations r(M ) = rp (M ) + 1 = r(A) + r(M/A) and rp (M ) = rp (A) + rp (M/A). We also have r(M/A) = rp (M/A). Otherwise, M/A is a free module and M = A ⊕ C with C ∼ = M/A (Theorem 5.4 (Part I)), which contradicts the property that M is a coreduced module. By using the above relations, it is easy to verify that r(A) = rp (A). Therefore, A is free. (3) =⇒ (1). We assume that A is a pure submodule in M with A = M and there exists a nontrivial decomposition M/A = X/A ⊕ Y /A. We have M/X ∼ = (M/A)/(X/A) ∼ = Y /A, where Y /A is a free module, since Y is a finitely generated free module. This leads to the decomposition M = X ⊕ Z, where Z is a free module. This contradicts the property that M is a coreduced module. Consequently, M is a copurely indecomposable module. The equivalence of (2) and (4) can be proved by using the duality F and the equivalence of properties (1) and (3) of Proposition 25.1. (3) =⇒ (5). Let ϕ : N → M be a nonzero homomorphism, where N is a coreduced torsion-free module. If r(Im(ϕ)) = r(M ), then Im(ϕ) and N/ Ker(ϕ) are free modules. Consequently, N = Ker(ϕ) ⊕ G, where G is a free module. This contradicts the property that N is a coreduced module. Therefore, r(Im(ϕ)) = r(M ). We also have that Im(ϕ) is a coreduced module (the proof of (3) =⇒ (1) has a similar point). Thus, we have the following situation. There is a submodule A of the module M such that r(A) = r(M ) and A is a coreduced module. We need to prove that M/A is a bounded module. Let X be a free submodule of the module A of maximal rank. It follows from Proposition 23.1 and (2) that M/X = C ⊕ D, where C is a direct sum of a finite number of cyclic primary modules and D is the cocyclic module R(p∞ ). Similarly, A/X = C1 ⊕ D1 . Here, D1 = 0; otherwise, pk A ⊆ X for some k and A is a free module. Consequently, D1 = D. We can also assume that C1 ⊆ C. In this case, M/A ∼ = C/C1 and M/A is a bounded module. (5) =⇒ (3). We assume that A is a nonzero submodule of the module M such that r(A) < r(M ) and A is not free. Then A = N ⊕ X, where N is a coreduced module and either X is a free module or X = 0. The existence of the embedding N → M contradicts (5). Consequently, (3) holds. Thus, purely indecomposable modules and copurely indecomposable modules correspond to each other under the Arnold duality. Consequently, if we know properties of purely (resp., copurely) indecomposable 3268
modules, then we can obtain properties of copurely (resp., purely) indecomposable modules by using the duality. Using Proposition 25.2, Proposition 25.5, and the functor F , we obtain the following result. Corollary 25.6. For any two copurely indecomposable modules M and N , the following conditions are equivalent: (1) M ∼ N ; (2) Hom(M, N ) = 0 and Hom(N, M ) = 0; (3) r(M ) = r(N ) and Hom(M, N ) = 0. The information about the endomorphism ring of a copurely indecomposable module M is considerably less than the information about the endomorphism ring of a purely indecomposable module. Since F is a duality, the rings K End(M ) and K End(F (M )) are anti-isomorphic to each other (the anti-isomorphism is defined in remarks to Chapter 7). However, F (M ) is a purely indecomposable module and K End(F (M )) is a field (Proposition 25.3 and Proposition 25.5). Consequently, we have an anti-isomorphism. Therefore, End(M ) is a commutative domain and K End(M ) is a field that is isomorphic to some field P , where and the extension degree [P : K] is finite (see Proposition 25.3). K⊆P ⊆K A torsion-free module that is quasi-isomorphic to a copurely indecomposable module is copurely indecomposable. For example, this follows from Proposition 25.5. Consequently, copurely indecomposable modules can be classified up to quasi-isomorphism by using the Arnold duality and Theorem 25.4. Two quasiisomorphic copurely indecomposable modules are not necessarily isomorphic to one another (Example 25.7). The classification of copurely indecomposable modules up to isomorphism is a considerably more difficult problem. In the remaining part of this section, we partially study this problem. The elements of a copurely indecomposable module of rank n + 1, n ≥ 1, can be represented by vectors n (the vectors are elements of the free R-module n of rank n). We call attention to the property in (R) (R) n that all subsequent events occur in the K-space (K) of dimension n. In particular, all considered modules n and K n ) are contained in this space. and spaces (for example, (R) then the symbol A[Γ] denotes the pure R-submodule in (R) n generated by Rn and If a1 , . . . , an ∈ R, n . It is easy to prove that A[Γ] is a coreduced module iff the elements the vector Γ = (a1 , . . . , an ) from (R) 1, a1 , . . . , an are linearly independent over R. In this case, A[Γ] is a copurely indecomposable module of p-rank n and the rank of A[Γ] is equal to n + 1 (see Exercise 8). Now let M be a copurely indecomposable module of rank n + 1 with basis submodule B of rank n. where B is a free We note that n = rp (M ). The module M is a pure submodule of the completion B, n and B R-module of rank n. Clearly, we can assume that M is isomorphic to some pure submodule in (R) n such that M is isomorphic is isomorphic to R . Since r(M ) = n + 1, there exist elements a1 , . . . , an in R to the module A[Γ], where Γ = (a1 , . . . , an ). Therefore, the formulated classification problem is reduced to the description of copurely indecomposable modules of the form A[Γ]. An essentially equivalent method of constructing copurely indecomposable modules is presented in Exercise 7. We present several simple properties of the copurely indecomposable modules A[Γ] and their endomorphism rings. It is clear that Rn is a basis submodule of the module A[Γ]. Furthermore, we have the relations KA[Γ] = K n ⊕ KΓ (where KΓ is the K-subspace generated by the vector Γ) and n. A[Γ] = (K n ⊕ KΓ) ∩ (R) (Here, we use our agreement that all considered objects are contained in the same K-space.) It follows from the method for constructing of the functor F that F (A(a1 , . . . , an )) ∼ A[Γ] and F (A[Γ]) ∼ = A(a1 , . . . , an ). The rank of the purely indecomposable module A(a1 , . . . , an ) is equal to n + 1. The quasi-endomorphism algebra K End(A[Γ]) is isomorphic to the K-algebra K End(A(a1 , . . . , an )) contained in K ⊕ Ka1 ⊕ · · · ⊕ Kan (see the remark before Theorem 25.4). Consequently, End(A[Γ]) is a domain that is isomorphic to 3269
(some information on this subfield is contained some subring of the subfield K End(A(a1 , . . . , an )) in K in Proposition 25.3). that is algebraically independent over Example 25.7 (Arnold–Dugas [12]). Let {a1 , a2 } be a subset in R K (an algebraic independence is defined in Lemma 19.9 (Part I)), Γ = (a1 , a2 ), and Δ = (pa1 , a2 ). Then A[Γ] and A[Δ] are quasiisomorphic copurely indecomposable modules that are not isomorphic to one another. Proof. We have
2, A[Γ] = (K 2 ⊕ K(a1 , a2 )) ∩ (R) 2, A[Δ] = (K 2 ⊕ K(pa1 , a2 )) ∩ (R)
and
F (A[Γ]) ∼ = A(a1 , a2 ) = A(pa1 , a2 ) ∼ = F (A[Δ]). Consequently, A[Γ] and A[Δ] are quasi-isomorphic to one another. Denote by ϕ the endomorphism (p, 1) ⊕K such that ϕ coincides with the multiplication by p on the first summand and ϕ coincides of the space K with the identity mapping on the second summand. In a similar manner, we define the endomorphism 2 and ϕ(KA[Γ]) ⊆ KA[Δ]. Therefore, ϕ ∈ Hom(A[Γ], A[Δ]). Similarly, we 2 ⊆ (R) ψ = (1, p). Then ϕ(R) have ψ ∈ Hom(A[Δ], A[Γ]) and ϕψ = p(1, 1) = ψϕ. Since F is a duality and A(a1 , a2 ) = A(pa1 , a2 ), we have K Hom(A[Γ], A[Δ]) ∼ = K End(A(a1 , a2 )). However, K End(A(a1 , a2 )) = {r ∈ K ⊕ Ka1 ⊕ Ka2 | ra1 , ra2 ∈ K ⊕ Ka1 ⊕ Ka2 } (see the paragraph before Theorem 25.4). Considering this property and the algebraic independence of the elements a1 and a2 , we have K End(A(a1 , a2 )) = K. Consequently, Hom(A[Γ], A[Δ]) = Rϕ, since ϕ is not divisible by p; similarly, we have Hom(A[Δ], A[Γ]) = Rψ. If we assume that A[Γ] ∼ = A[Δ], then ϕ and ψ are isomorphisms, since they are not divisible by p. Consequently, modules A[Γ] and A[Δ] are not isomorphic to one another. We specialize Proposition 2.4 (Part I) on the representation of endomorphisms by matrices. Let we n have a direct sum of modules M = Ai . If a ∈ M and a = a1 + · · · + an (ai ∈ Ai ), then we represent i=1
the element a (if necessary) as a vector (a1 , . . . , an ). Let α ∈ End(M ) and let (αij ) be the matrix corresponding to α, where αij ∈ Hom(Ai , Aj ). By using the proof of Proposition 2.4 (Part I), it is easy to verify the matrix relation αa = a(αij ), where a = (a1 , . . . , an ) and the matrices (αij ) are multiplied by the usual rule of matrix multiplication and αa is the corresponding vector. In what follows, the matrices considered are n × n matrices over the ring R. Every matrix considered is invertible iff the determinant of the matrix is an invertible element of R. The following lemma contains a simple isomorphism criterion of modules of the form A[Γ]. It was already mentioned that matrices are multiplied according to the usual rule. Lemma 25.8. We assume that M = A[Γ] and N = A[Δ] are copurely indecomposable modules of rank n + 1. If Δ = sΓV + Λ for some nonzero element s ∈ K, an invertible matrix V , and the vector Λ from K n , then M and N are isomorphic to one another. n. n such that ϕ(x) = xV −1 for each vector x in (K) Proof. We take the automorphism ϕ of the space (K) n n It is clear that ϕ also is an automorphism of the module (R) . Furthermore, we have KA[Δ] = K ⊕ KΔ, (K n )V −1 = K n , ΔV −1 = sΓ + ΛV −1 ∈ KΓ ⊕ K n = KA[Γ], and ϕ(KA[Δ]) ⊆ KA[Γ]. n , ϕ(A[Δ]) ⊆ A[Γ]. We can assert that ϕ : A[Δ] → n and A[Γ] = KA[Γ] ∩ (R) Since A[Δ] = KA[Δ] ∩ (R) A[Γ] is an isomorphism, since ϕ(A[Δ]) is a pure submodule in A[Γ]. 3270
The meaning of the next lemma is as follows: if the modules A[Γ] and A[Δ] are quasiisomorphic to one another, then the vectors Γ and Δ are related to each other in some way. Lemma 25.9 (Arnold–Dugas [12]). Let M = A[Γ], r(M ) = n + 1, and let N be a torsion-free module that is quasiisomorphic to M . Then (1) N is isomorphic to the module A[ΓW ] for some upper triangular matrix W = (wij ) with nonzero determinant and h(wii ) > h(wij ) in each case where 1 ≤ i < j ≤ n and wij = 0; (2) N is isomorphic to the module A[ΓV D] for some invertible matrix V and a diagonal matrix D with diagonal elements pe(1) , . . . , pe(n) and e(1) ≥ · · · ≥ e(n); n such that M is isomorphic to A[Δ] and N is isomorphic to A[ΔD] (3) there exists a vector Δ ∈ (R) for some diagonal matrix D with diagonal elements pe(1) , . . . , pe(n) and e(1) ≥ · · · ≥ e(n). Proof. (1) Since M and N are quasi-isomorphic to one another, there exists a submodule L of the module M such that N is isomorphic to L and KL = KM = K n ⊕ KΓ (Corollary 24.4). First, we prove that L is isomorphic to A[ΓV ] for some matrix V with nonzero determinant. We note that Rn is a basis submodule of the module M and B = L ∩ Rn is a basis submodule of the module L with KB = K n (Exercise 3). We take a K-automorphism of the space K n that extends some isomorphism from B onto Rn ; we also take the corresponding invertible n × n matrix V over K. We choose pk (k ≥ 0) such that pk V = V is a → (pk R) n by matrix over R. It is clear that {bV | b ∈ B} = pk Rn . This relation allows us to define ϕ : B the relation ϕ(x) = xV . Here, ϕ is an R-isomorphism and ϕ : KL = K n ⊕ KΓ → KA[ΓV ] = K n ⊕ KΓV. Since
∩ KL and A[ΓV ] = (R) n ∩ (K n ⊕ KΓV ), L=B we have that ϕ is an isomorphism from L onto pk A[ΓV ]. Then (1/pk )ϕ is an isomorphism from L onto A[ΓV ], which is what we were required to prove. Now we can apply the standard procedure to the matrix V . Precisely, we premultiply the matrix V by some invertible matrices and obtain the required upper triangular matrix W (Exercise 1). By Lemma 25.8, such multiplications do not change the isomorphism class of the module A[ΓV ]. Thus, A[ΓV ] is isomorphic to A[ΓW ]. Since N is isomorphic to A[ΓV ], the proof of (1) is completed. (2) By (1), N is isomorphic to A[ΓW ] for some matrix W with nonzero determinant. Since R is a discrete valuation domain, there exist invertible matrices V and U with V −1 W U = D, where D is a diagonal matrix with diagonal elements pe(1) , . . . , pe(n) and e(1) ≥ · · · ≥ e(n) (Exercise 2). Then W U = V D, and it follows from Lemma 25.8 that A[ΓW ] is isomorphic to A[ΓW U ] = A[ΓV D]. (3) We set Δ = ΓV , where V is the matrix from (2). Then the module M = A[Γ] is isomorphic to A[Δ] by Lemma 25.8. Now we use (2) and obtain that N is isomorphic to A[ΔD]. Lemma 25.9 reduces the isomorphism problem for two quasi-isomorphic copurely indecomposable modules M and N to the case where M = A[Γ] and N = A[ΓD] for some diagonal matrix D with diagonal elements pe(1) , . . . , pe(n) and e(1) ≥ · · · ≥ e(n). We present an isomorphism criterion for copurely indecomposable modules M with r(End(M )) = 1. : K] is In this case, End(M ) = R. There are many such modules provided the extension degree [K sufficiently large (see Exercise 10). Theorem 25.10 (Arnold–Dugas [12]). We assume that M = A[Γ] is a copurely indecomposable module with r(End(M )) = 1 and N = A[ΓD] for some diagonal matrix D with diagonal elements pe(1) , . . . , pe(n) and e(1) ≥ · · · ≥ e(n). Then M and N are isomorphic to one another iff e(1) = · · · = e(n). n → (R) n by ϕ(x) = xD. Then ϕ induces a homomorphism from Proof. We define a mapping ϕ : (R) M = A[Γ] into N = A[ΓD], since ϕ : KM = K n ⊕ KΓ → KN = K n ⊕ KΓD. 3271
The height of the homomorphism ϕ (as an element of the R-module Hom(M, N )) is equal to e(n). Furthermore, K Hom(M, N ) ∼ = K End(M ) = K. Consequently, Hom(M, N ) = R(1/pe(n) )ϕ, since the homomorphism (1/pe(n) )ϕ is not divisible by p. Consequently, the homomorphism is a generator of the R-module Hom(M, N ). Therefore, it is clear that M and N are isomorphic to one another iff (1/pe(n) )ϕ is an isomorphism. Since (1/pe(n) )ϕ coincides with the multiplication by the matrix (1/pe(n) )D, the modules M and N are isomorphic to one another iff this matrix is invertible. This is equivalent to the existence of the relations e(1) = · · · = e(n). In [12], Arnold and Dugas also study the classification problem for copurely indecomposable modules M such that r(End(M )) = r(M ) or 1 < r(End(M )) < r(M ). We note that r(End(M )) ≤ r(M ), since End(M ) is a domain. Indeed, for a fixed nonzero element a ∈ M , the mapping α → α(a) (α ∈ End(M )) is an R-module embedding End(M ) → M . The cited paper contains interesting examples of copurely indecomposable modules of rank 3. They demonstrate obstructions to the search for an exact description of isomorphism classes of copurely indecomposable modules that are quasi-isomorphic to the given copurely indecomposable module. Arnold, Dugas, and Rangaswamy [14] consider finite direct sums of purely indecomposable modules. In [14], such sums are called pi-decomposable modules. In this paper, the following topics are studied: the search for number invariants with respect to isomorphism of pidecomposable modules, torsion-free homomorphic images of pi-decomposable modules, and modules that are quasi-isomorphic to pi-decomposable modules. Exercise 1. Let V be an n × n matrix over a ring R with nonzero determinant. Prove that we can premultiply the matrix V by some invertible matrices and obtain an upper triangular matrix W = (wij ) with nonzero determinant, and h(wii ) > h(wij ) in each case where 1 ≤ i < j ≤ n and wij = 0. (The multiplication of a column by an invertible element and interchanging columns can be obtained by indicated multiplications.) Exercise 2. If W is a matrix with nonzero determinant, then there exist invertible matrices V and U such that V −1 W U is a diagonal matrix with diagonal elements pe(1) , . . . , pe(n) and e(1) ≥ · · · ≥ e(n). Exercise 3. Let B be a basis submodule of a torsion-free module M of finite rank. If M ⊂ M and M/M is a bounded module, then B = B ∩ M is a basis submodule of the module M . Furthermore, if B ⊂ B and B/B is bounded, then there exists an M ⊂ M such that M/M is a bounded module and B = B ∩ M . Exercise 4. If M is a copurely indecomposable module and A is a pure submodule in M , then the factor module M/A is a strongly indecomposable module. Exercise 5. (a) Let M be a reduced torsion-free module such that rp (M ) = n and r(M ) = n + k. Then there exists an exact sequence of modules, 0 → M → A1 ⊕ · · · ⊕ An → D → 0, where D is a divisible torsion-free module and Ai is a purely indecomposable module such that r(Ai ) ≤ k + 1 for i = 1, . . . , n. (b) Formulate and prove the dual assertion for a coreduced module M . Exercise 6. Verify the relations A(a1 , . . . , an ) = KA(a1 , . . . , an ) ∩ R, KA(a1 , . . . , an ) = K ⊕ Ka1 ⊕ · · · ⊕ Kan , and K End(A(a1 , . . . , an )) = {r ∈ K ⊕ Ka1 ⊕ · · · ⊕ Kan | rai ∈ K ⊕ Ka1 ⊕ · · · ⊕ Kan , 1 ≤ i ≤ n}. 3272
Exercise 7 (Kaplansky [160] and Arnold [12]). Let V be a vector K-space of dimension n + 1 with basis Let aj = lim aij , where aij ∈ R, j = 1, . . . , n, x, y1 , . . . , yn . Choose some elements a1 , . . . , an ∈ R. i = 0, 1, 2, . . .. We set n wi = x + aij yj , i ≥ 0. j=1
Let M be the R-submodule in V generated by the elements y1 , . . . , yn , w0 , w1 /p, . . . , wi /pi , . . .. Then (1) rp (M ) = n, r(M ) = n+1, and the elements y1 , . . . , yn generate a basis submodule of the module M ; (2) M is a coreduced module iff the system {1, a1 , . . . , an } is linearly independent; in this case, M is a copurely indecomposable module; (3) every module M such that rp (M ) = n and r(M ) = n+1 can be constructed (up to an isomorphism) by the given method. Exercise 8. Prove that the module A[Γ] is copurely indecomposable iff {1, a1 , . . . , an } is a linearly inde pendent system of elements of the R-module R. Exercise 9. Prove that F (A(a1 , . . . , an )) ∼ A[Γ] and F (A[Γ]) ∼ = A(a1 , . . . , an ). Exercise 10. Let Γ be a vector (a1 , . . . , an ) such that {1, a1 , . . . , an , a1 ai , . . . , an ai } is a linearly independent system for every i. Then r(End(A[Γ])) = 1. Exercise 11. Let M ⊆ N , where M is a purely indecomposable module and N is a reduced torsion-free module of finite rank. Then M ∼ = M∗ , where M∗ is the pure submodule in N generated by M (see property (7) in Sec. 7, Part I). 26.
Indecomposable Modules over Nagata Valuation Domains
In this section, we consider the classification problem of indecomposable torsion-free modules of finite rank. In full generality, the problem is very difficult (some information on this topic is contained in remarks at the end of the section and in remarks at the end of the chapter). We call attention to the property that purely indecomposable and copurely indecomposable modules are always indecomposable. We consider this problem for modules over Nagata valuation domains. For such discrete valuation domains, the field has several special properties. We use some of these properties. First, we consider the extension K < K properties and present several general results on field extensions. All of them are well known and are contained in the works of Bourbaki [34] and Zariski and Samuel [315, 316]. The reader certainly knows the basic notions of the theory of field extensions. We only mention separable and purely inseparable extensions and properties of valuations in fields. Let F be a field. A polynomial f (x) over the field F is said to be separable if f (x) has no multiple roots in any extension of the field F . We assume that the field E is an algebraic (in particular, finite) extension of the field F . An element a of the field E is said to be separable over F if a is a root of a polynomial that is separable over F . This is equivalent to the property that the minimal polynomial of an element a over the field F is separable. If every element of E is separable over F , then we say that E is a separable extension of the field F . An element a is said to be purely inseparable over F if there exists n an integer n ≥ 0 such that aq is contained in F for some prime integer q. If every element of E is purely inseparable over F , then we say that E is a purely inseparable extension of the field F . In this case, the characteristic of the field F is necessarily finite, say, q, and the minimal polynomial of every element of n E over F has the form xq − c for some n ≥ 0 and c ∈ F . If E is an arbitrary algebraic extension of the field F , then all F -separable elements of the field E form a field that is a maximal separable extension of the field F contained in E. We define several new notions related to valuations in fields and the corresponding valuation rings (valuations and valuation rings were defined in Sec. 3 (Part I)). Let F be a field, (Γ, ≤) be a linearly ordered group, and ν be a valuation of the field F with values in Γ. The image of ν under the mapping 3273
F ∗ → Γ is an ordered subgroup in Γ which is called the value group of the valuation ν. Let ν1 and ν2 be two valuations of the field F with values in Γ. They are said to be equivalent if there exists an order-preserving isomorphism ϕ from the value group ν1 F ∗ onto the value group ν2 F ∗ (i.e., ϕ is an isomorphism of ordered groups) such that ν2 = ν1 ϕ. The valuatuons ν1 and ν2 are equivalent iff the corresponding valuation rings, considered as subrings of the field F , coincide. Therefore, there exist a bijective correspondence between valuation rings in the field F and equivalence classes of valuations. We briefly recall several results about the behavior of valuations under field extensions. Let E be an extension of a field F and let R be the valuation ring for ν in the field E (with values in Γ). The restriction ν of the mapping ν to F is a valuation of the field F with valuation ring R that is equal to F ∩R . The value group of the valuation ν is a subgroup in the value group of the valuation ν . Conversely, every valuation of the field F has an extension to a valuation of the field E. The index [ν E ∗ : νF ∗ ] of the value group of the valuation ν in the value group of the valuation ν is called the ramification index of the valuation ν with respect to the valuation ν; the index is denoted by e(ν /ν). From the viewpoint of Sec. 3 (Part I), the rings R and R are local. The fields R/J(R) and R /J(R ) are called residue fields of the rings R and R , respectively. Since J(R) = J(R ) ∩ F , we have the field embedding R/J(R) → R /J(R ) (r + J(R) → r + J(R ), r ∈ R). By using this embedding, the residue field R/J(R) is identified with some subfield of the residue field R /J(R ). The extension degree [R /J(R ) : R/J(R)] of the field R /J(R ) with respect to the field R/J(R) is called the residue degree of the valuation ν with respect to the valuation ν; it is denoted by f (ν /ν). If (νi )i∈I is a family of valuations of the field E extending the valuation ν and every valuation of the field E extending ν is equivalent to exactly one of the valuations νi , then (νi )i∈I is called a complete extension system of the valuation ν on the field E. Important properties of extensions of valuations are contained in the following theorem (see Bourbaki [34, Chapter VI, Sec. 8]). Theorem. Let F be a field, ν be the valuation of the field F , and E be a finite extension of degree n of the field F . Then (a) every complete system (νi )i∈I of extensions of the valuation ν on E is finite; (b) the inequality e(νi /ν)f (νi /ν) ≤ n holds; i∈I
(c) the valuation rings for νi are pairwise incomparable with respect to inclusion. In some cases, the inequality in (b) can be replaced by an equality. Corollary. If the valuation ν is discrete and the extension E is separable over F , then s e(νi /νi )f (νi /ν) = n, where (νi )1≤i≤s is a complete system of extensions of the valuation ν to the i=1
field E. In connection with discrete valuations, we also note that if ν is some extension of the valuation ν of the field F to the field E, then the valuation ν is discrete iff the valuation ν is discrete. Let Vγ = {x ∈ F | ν(x) > γ}, where γ ∈ Γ. The set of all subgroups Vγ (γ ∈ Γ) of the additive group of the field F forms a basis of neighborhoods of zero of the topology Tν in the field F ; this topology is called the topology defined by the valuation ν. The topology Tν is Hausdorff, and the mapping ν : F ∗ → Γ is continuous if we assume that the group Γ is equipped with the discrete topology. If ν is discrete, then the topology on the valuation ring for ν induced by the topology Tν coincides with the p-adic topology. (However, this is not true in the general case (Exercise 2).) The completion F of the field F with respect to the topology Tν is a topological field; the valuation ν is extended to a unique valuation ν : F∗ → Γ by continuity. The topology of the field F is the topology defined by the valuation ν. The valuation ring for 3274
ν is the completion of the valuation ring for ν. For a discrete valuation ν, the completion of F coincides with the field of fractions of the valuation ring ν. are topological fields with respect to the Let R be a discrete valuation domain. Then K and K corresponding discrete valuation ν, K is the completion of the field K, where K is still the field of is the field of fractions of the p-adic completion R of the domain R. fractions of the domain R and K The p-adic topology on R or R coincides with the induced topology. The valuatuon ν of the field K has a unique extension to a valuation of the field K. ∩ K, it is easy to verify that the dimension Now we define Nagata valuation domains. Since R = R of the K-space K is equal to the rank of the R-module R. The domain R is called a Nagata valuation : K] of the field K over K is finite and exceeds 1 (therefore R is not a domain if the extension degree [K complete domain). The existence of such valuation domains was proved by Nagata [236, Example 3.3]. is a purely inseparable extension of the field Let R be a Nagata valuation domain. We prove that K K (see Ribenboim [246]). Let p be an infinite cyclic group, where p is the prime element of the ring R, and let ν : K ∗ → p be the canonical valuation mentioned in Sec. 3 (Part I). The valuation can be ∗ → p. It was mentioned earlier that ν is a unique extension extended to the canonical valuation ν : K of ν to a valuation of the field K. Every valuation of Now we assume that F is a separable extension of the field K contained in K. Consequently, ν has a unique extension to a the field F can be extended to a valuation of the field K. valuation of the field F , and s = 1 in the formula from the above corollary. Let ν be an extension of ν to a valuation of the field F . Then e(ν /ν)f (ν /ν) = [F : K]. It follows from the choice of ν that e(ν /ν) = 1. Furthermore, since R = R/pR, R/pR ⊆ R /pR ⊆ R/p where R is the valuation ring ν (see the point before the definition of the residue degree of the valuation), is we have R/pR = R /pR , whence f (ν /ν) = 1. We obtain [F : K] = 1 and F = K; this implies that K purely inseparable over K. We can draw an important conclusion that the fields K and K are of finite : K] is a power of q. For every a ∈ K, there exists an integer n ≥ 0 with aqn ∈ K. characteristic q and [K Before passing to modules over Nagata valuation domains, let us make some remarks. First, if the degree : K] is infinite, then the rank of the R-module R is infinite. In this case, there exist indecomposable [K : K] = 1, then R = R. In this torsion-free modules of any finite rank (Example 11.9 (Part I)). If [K case, every indecomposable torsion-free module is equal to R or K (Corollary 11.8 (Part I)). In what follows, the results of the previous section will be substantially used. We especially note that all reduced indecomposable modules appearing below are purely indecomposable or copurely indecomposable. Up to the end of the section, R is a Nagata valuation domain. We begin with the minimally possible = 1. Theorem 26.1 and : K] = 2. For such domains R, we have r(R) = 2 and rp (R) case, where [K Theorem 26.2 were obtained by Zanardo [314], who used the matrix Kurosh invariants for torsion-free modules of finite rank (see also remarks to the chapter in connection with these invariants). Theorem 26.1. Let M be an indecomposable module of finite rank. Then M is isomorphic to one of the modules R, K, and R. Proof. If r(M ) = 1, then M is isomorphic to R or K. Let r(M ) = 2. For rp (M ) = 2, M is a free module (see the beginning of Sec. 23); for rp (M ) = 0, M is divisible. In any case, M is a decomposable module, which is impossible. Therefore, rp (M ) = 1. Consequently, M is a purely indecomposable module (purely indecomposable and copurely indecomposable modules were studied in the previous section). It Since r(N ) = r(R), we have N = R, whence is isomorphic to some pure submodule N of the module R. ∼ M = R. 3275
We assume that r(M ) = 3. Similarly to the previous paragraph, we have that either rp (M ) = 1 or which contradicts the rp (M ) = 2. In the first case, M must be isomorphic to some pure submodule in R, conditions for ranks. In the case rp (M ) = 2, the module M is copurely indecomposable. Then F (M ) is a purely indecomposable module of rank 3 (Proposition 25.5). As above, this is impossible. Thus, there are no indecomposable modules of rank 3. Now we assume that every reduced torsion-free module of We take a reduced torsionrank n is decomposable; fr(R) = ∞ means that for any positive integer m, there exists an indecomposable torsion-free R-module whose rank is finite and exceeds m. Matlis studied domains with fr(R) = 1. If R is a valuation domain, then by a result of Kaplansky [162] and Matlis [213], we have that fr(R) = 1 iff R is a maximal domain. Now we can represent the results that are proved or mentioned in Sec. 26 as : K] is infinite, then fr(R) = ∞ and fr(R) = n for [K : K] = n, where follows. If the extension degree [K n = 1, 2, 3. If [K : K]≥4, then fr(R) = ∞. The uniqueness of a direct decomposition of a given module usually means that the Krull–Schmidt theorem holds for the decomposition. The end of Sec. 27 contains some information about this theorem and related studies. V´ amos [284] and Lady [186] have proved that the Krull–Schmidt theorem holds for every torsion-free module of finite rank over a discrete valuation domain iff the domain is a Henselian ring. A general theory of Henselian rings was developed in the work of Nagata [236]. At the end of Sec. 25, the results of Arnold, Dugas, and Rangaswamy on π-decomposable modules were mentioned. A module is said to be π-decomposable if it is a finite direct sum of purely indecomposable
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modules. The class of π-decomposable modules can be enlarged in the following way. We denote by A the class of all finite direct sums of torsion-free modules M of finite rank such that End(M ) is a discrete valuation domain (see, for example, Exercise 7 in Sec. 37); A is the class of all modules that are quasi-isomorphic to modules in A. Problem 11. Study properties of modules in the classes A and A . Problem 12. (a) Describe the Jacobson radical of the endomorphism ring of a module in the class A . (b) Describe automorphism groups of modules in A . In the case of Abelian groups, the modules in the class A correspond to finite direct sums of torsion-free groups A of finite rank such that End(A) is a strongly homogeneous ring. (Facts related to similar groups are presented in [177, Chapter 7] and in exercises from [177, Sec. 41].) The π-decomposable modules correspond to direct sums of torsion-free groups of finite rank with cyclic p-basis subgroups (such groups are considered in [177, Sec. 44]). All almost completely decomposable groups (i.e., subgroups of finite index in completely decomposable torsion-free groups of finite rank) are contained in an analog of the class A . An explicit theory of these groups is presented in Mader’s work [210].
Chapter 6 MIXED MODULES In Chapter 6, we consider the following topics: the uniqueness and refinements of decompositions in additive categories (Sec. 27); isotype, nice, and balanced submodules (Sec. 28); the categories Walk and Warf (Sec. 29); simply presented modules (Sec. 30); decomposition bases and extending of homomorphisms (Sec. 31); Warfield modules (Sec. 32). In Secs. 11, 21, and 22 (Part I), we already dealt with mixed modules. This chapter is completely devoted to such modules. At times, the theory of mixed modules was in stagnation as compared with the theories of primary modules and torsion-free modules. At present, the theory is being intensively studied. This situation is already obvious in many studied subclasses of direct sums of mixed modules of torsion-free rank 1. Mixed modules “are assembled” from primary modules and torsion-free modules. Consequently, they inherit the complexity of the structure of these objects. The “assembling” process also increases the difficulties. According to the definition presented in Sec. 4 (Part I), a mixed module M necessarily contains nonzero elements of finite order and elements of infinite order. We recall that t(M ) is the torsion submodule (or the primary submodule) of the module M , i.e., the set of all elements in M of finite order. The rank of the torsion-free factor module M/t(M ) was called earlier the torsion-free rank of the module M . In this chapter, the torsion-free rank of the mixed module M is called the rank of the module M for brevity. This does not lead to any confusion. A mixed module M is said to be split if M = t(M ) ⊕ F for some torsion-free module F . The module M is always split provided t(M ) is a bounded module or M/t(M ) is a free module. The chapter is very rich in content. In Sec. 27, we dealt with an arbitrary additive category. In Sec. 29, we dealt with two categories of module origin. We consider several types of submodules playing an important role for mixed modules (Sec. 28). In the remaining sections (Secs. 30–32), the theory of one class of mixed modules is developed. Theorem 32.6 is a culmination of the theory. Without loss of generality, we assume that all modules are reduced (unless otherwise specified). This considerably simplifies the proofs. 3279
The symbol ω always denotes the first infinite ordinal number. 27.
Uniqueness and Refinements of Decompositions in Additive Categories
We consider two sufficiently general isomorphism theorems for direct decompositions and refinements of direct sums in an additive category. (These notions were introduced before Theorem 8.6 (Part I).) The section is based on the paper of Walker and Warfield [290]. We work in some additive category E. Section 1 (Part I) and Sec. 24 contain some information on additive categories. We also recall several simple category-theoretical notions. Other more special notions will be defined if necessary. A morphism f : A → B of the category E is called a monomorphism if for every object X and any morphisms g, h : X → A, the relation gf = hf implies the relation g = h. The composition of two monomorphisms is a monomorphism; if f g is a monomorphism, then f is a monomorphism. If f : A → M is a monomorphism, then A is called a subobject of the object M ; we write A ⊂ M in this case. Strictly speaking, subobjects coincide with pairs (A, f ). To simplify the presentation, every-where, we will write A instead of (A, f ) or f : A → B if this does not lead to confusion. In this case, we are basing this on the notion of an equivalence for monomorphisms. Two monomorphisms f : A → M and g : B → M are said to be equivalent if there exist morphisms e : A → B and h : B → A such that f = eg and g = hf . It is easy to verify that such morphisms e and h are uniquely defined isomorphisms. Now we can choose a representative in every class of equivalent monomorphisms; we assume that the representative is a subobject. The relation ⊂ between subobjects is a partial order. The greatest lower bound of two subobjects A and B of the object M (if it exists) is called the intersection of A and B; it is denoted by A ∩ B. We pass to direct sums in the category E. Many properties of the sums are similar to the corresponding properties of direct sums of modules. We also say that a subobject A ⊂ M is a direct summand of the object M if there exists a subobject B ⊂ M such that M = A ⊕ B. We already have Lemma 24.1. We consider two additional properties. Lemma 27.1. Let M be an object of the category E, M = A ⊕ B, eA and eB be the corresponding embeddings, and pA and pB be the corresponding projections. Then (1) if eC : C → M is a monomorphism and eC pA : C → A is an isomorphism, then M = C ⊕ B with embeddings eC and eB and projections qC = pA (eC pA )−1 and qB = (1M − qC eC )pB ; (2) if M = C ⊕ D with embeddings eC and eD and projections pC and pD , and if eA pC : A → C is an isomorphism, then B ∼ = D. Proof. (1) It follows from the calculations that eC qB = 0, eB qC = 0, eC qC = 1C , eB qB = 1B , and 1M = qC eC + qB eB . It remains to use to Lemma 24.1. (2) By (1), we obtain a decomposition M = A ⊕ D with embeddings eA and eD and projections qA and qD , where qA = pC (eA pC )−1 and qD = (1M − qA eA )pD . We claim that eD pB : D → B is an isomorphism with the inverse isomorphism eB qD : B → D. Indeed, we have eB qD eD pB = eB (qA eA + qD eD )pB = eB 1M pB = eB pB = 1B and eD pB eB qD = eD (pA eA + pB eB )qD = eD qD = 1D . The definitions of isomorphic direct decompositions of a module and refinements of a direct decomposition, which are presented before Theorem 8.6 (Part I), can be transferred to direct decompositions of objects of an additive category without changes. First, we consider finite direct sums. Theorem 27.2 (Walker–Warfield [290]). Let M be an object of the additive category E such that M = n m Ai , where End(Ai ) is a local ring for all i. If M = Bj , where the ring End(Bj ) has no nontrivial
i=1
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j=1
idempotents for j = 1, . . . , m, then m = n and there exists a permutation θ of the set {1, 2, . . . , n} such that Ai ∼ = Bθ(i) for i = 1, . . . , n. An arbitrary direct decomposition of the object M can be refined to a decomposition consisting of at most n indecomposable summands. Proof. Let ei : Ai → M and fj : Bj → M be the embeddings, and let pi : M → Ai and qj : M → Bj be m e1 qj fj p1 is the identity morphism of the object A1 . the projections for the given two direct sums. Then j=1
Since End(A1 ) is a local ring, one of the summands is an invertible element of this ring (Proposition 3.1 (Part I)). We can assume (after renumbering) that the summand is the element γ = e1 q1 f1 p1 . Now we consider the endomorphism f1 p1 γ −1 e1 q1 of the object B1 . After the calculation, we obtain that the endomorphism is an idempotent. By assumption, the endomorphism is equal to 0 or 1. It is easy to verify that (e1 q1 f1 p1 γ −1 )2 is the identity morphism of the object A1 . Consequently, the factor f1 p1 γ −1 e1 q1 of it is equal to the identity morphism of the object B1 . We obtain that the morphism f1 p1 is an isomorphism from B1 onto A1 with the inverse isomorphism γ −1 e1 q1 . It follows from Lemma 27.1 (1) that M = B1 ⊕ A2 ⊕ · · · ⊕ An . By Part (2) of this lemma, we have A2 ⊕ · · · ⊕ An ∼ = B2 ⊕ · · · ⊕ Bm . The assertion is verified by using induction on the number of summands Ai . We pass to the second assertion. Let S be the endomorphism ring of the object M . The elements p1 e1 , . . . , pn en form a complete orthogonal system of idempotents of the ring S (see Lemma 24.1). Conn sequently, we have a decomposition of the ring S into the direct sum of right ideals S = (pi ei )S. i=1
Equivalently, we have the direct decomposition of the right S-module S. Now we have EndS ((pi ei )S) ∼ = (pi ei )S(pi ei ) ∼ = End(Ai ) (see Exercise 2 for the second isomorphism). This implies that for every i, the endomorphism ring EndS ((pi ei )S) is local. Now we take any another decomposition of the ring S into a direct sum of right ideals. We have two decompositions of the ring S, and then we can apply one of the variants of the Krull–Schmidt theorem (e.g., Theorem 24.2) to these decompositions. As a result, we obtain that any decomposition contains no more than n summands. Therefore, the required assertion on refinements follows from Lemma 24.1. We cannot say that endomorphism rings of indecomposable direct summands from the theorem have no nontrivial idempotents, since idempotents are not necessarily split. In particular, decompositions of the ring S need not induce decompositions of the object M . In general, the proved theorem is weaker than Theorem 24.2. In the category E, the direct sum of every infinite set of objects can be defined similarly to the definition of finite direct sums. For infinite direct sums, Lemma 24.1 holds in the necessity part (except for the relation 1A = q1 e1 + · · · + qn en ). If the category E has the direct sum of every infinite set of objects, then we say that E is a category with infinite sums. In such a category, many properties of direct sums of moduleshold. For example, we can “amalgamate” or “disintegrate” direct summands. For example, let M = Ai and let J ⊂ I. Then there exists a direct sum M = M1 ⊕ M2 , where M1 = Ai i∈I i∈J and M2 = Ai ; the embeddings and the projections of this direct sum are naturally related to the i∈I\J
embeddings and the projections of the original sum. We also formulate the known notion of the kernel of a morphism. The pair (L, l), where l : L → A is a monomorphism, is called the kernel of the morphism f : A → B if lf = 0 and every morphism g : X → A with gf = 0 can be uniquely represented in the form g = hl for some h : X → L. If there exist different kernels of the morphism f , then they define the same subobject of the object A in the sense indicated 3281
in the beginning of the section. We denote it by Ker(f ). In the category of modules, it is clear that we obtain the ordinary notion of the kernel of a homomorphism. We say that E is a category with kernels (or E has kernels) if every morphism of the category E has a kernel. Lemma 27.3. Let E be an additive category with kernels. The following assertions hold : (1) if π ∈ End(M ) is an idempotent, then M = Ker(π) ⊕ Ker(1 − π); (2) if A, C ⊂ M and A is the kernel, then the intersection A ∩ C exists; (3) if M = A ⊕ B and A ⊂ C ⊂ M , then C = A ⊕ (C ∩ B). Proof. (1) We repeatedly use the definition of a kernel. Let i : Ker(π) → M and j : Ker(1 − π) → M be monomorphisms. Then iπ = 0 and j(1 − π) = 0. Since (1 − π)π = 0, there exists a unique morphism p : M → Ker(π) with 1 − π = pi. Similarly, there exists a morphism q : M → Ker(1 − π) with π = qj. Then i and j and p and q satisfy the conditions of Lemma 24.1, i.e., i and j are the embeddings and p and q are the projections for the sum presented in (1). It is obvious that pi + qj = 1M . Furthermore, it follows from the relations i = 1 · i and i = (ip)i that ip = 1; similarly, iq = 1. Finally, it follows from the relations 0 · j = 0 and (iq)j = 0 that iq = 0. Similarly, we have jp = 0. (2) Let i : A → M be the kernel of the morphism f : M → N , and let j : C → M be a monomorphism. We take the kernel l : L → C of the morphism jf : C → N . We verify that the monomorphism lj : L → M is an intersection of the subobjects A and C. We have L ⊂ C. Since (lj)f = 0, there exists a morphism h : L → A with lj = hi. Therefore, h is a monomorphism and L ⊂ A. We assume that X ⊂ M , X ⊂ A, X ⊂ C, g, gA and gC being the corresponding monomorphisms. We can assume that gA i = g = gC j, since we have equivalent monomorphisms. It follows from gA (if ) = 0 that gC (jf ) = 0. Consequently, there exists a unique morphism k : X → L with gC = kl. Since the morphism k is a monomorphism, we have X ⊂ L. (We note that gA = kh.) Thus, L is the intersection A ∩ C. (3) Let eA : A → M , let pA : M → A be the embedding and the projection, and let k : A → C and eC : C → M be monomorphisms; we can assume that keC = eA . The morphism eC pA k is an idempotent of the ring End(C). Consequently, it follows from (1) that we have the decomposition C = Ker(1 − π) ⊕ Ker(π). Let l : C ∩ B → C be a monomorphism, where l is the kernel of the morphism eC pA (see (2)). Then Ker(1 − π) = A and Ker(π) = C ∩ B. In particular, Part (2) of the proved lemma is applicable in the case where A is an arbitrary direct summand of the module M , since any direct summand is the kernel of the corresponding projection. To present the main results, we need some notions related to properties of direct sums; they themselves are of great interest. We say that an object M of an additive category admits the exchange property if M satisfies the following condition: if the object M is a direct summand of some object A that is the direct sum of the subobjects Ai , i ∈ I, i.e., A=M ⊕N = Ai , i∈I
then there exist decompositions Ai = Ai ⊕ Bi such that A = M ⊕
i∈I
Ai . If this condition holds in any
case where I is a finite set, then M has the finite exchange property. If M = A ⊕ B with embeddings e1 and e2 and projections p1 and p2 and f : M → N is an isomorphism, then it is clear that we also have N = A ⊕ B with embeddings e1 f and e2 f and projections f −1 p1 and f −1 p2 . Lemma 27.4. If A ⊕ B ⊕ C = A ⊕ Di and B has the exchange property, then i∈I
A⊕B⊕C =A⊕B⊕
i∈I
where 3282
Di
⊂ Di , i ∈ I.
Di ,
Proof. Using Lemma 27.1 and the above remark, we have B ⊕ C = property. Therefore,
Di = B ⊕
i∈I
where
Di
i∈I
Di . The object B has the exchange
Di ,
i∈I
⊂ Di . Therefore, A⊕B⊕C =A⊕B⊕
Di ,
i∈I
which is what we were required to prove. Now it is easy to obtain the following corollary. Corollary 27.5. Any direct sum of finitely many objects with the exchange property has the exchange property. In terms of the endomorphism ring, we can consider the case where the finite exchange property holds. Proposition 27.6. Let E be an additive category with kernels, and let M be an object in E. If the endomorphism ring of M is local, then M has the finite exchange property. Proof. We assume that A = M ⊕ N =
n i=1
Ai . Let e : M → A and ei : Ai → A be the corresponding
embeddings and let p : A → M and pi : A → Ai be the corresponding projections. Then 1M =
n
epi ei p.
i=1
Since the endomorphism ring of M is local, we have that one of the summands is an invertible element of the ring End(M ); let γ = ep1 e1 p for definiteness. We set π = pγ −1 ep1 e1 : A → A. Then π 2 = π, and by Lemma 27.3, A = Ker(π) ⊕ Ker(1 − π). (∗) Let k : Ker(1 − π) → A be an embedding. Since 0 = k(1 − π), we have k = kπ. Furthermore, it follows from the relation πpj = 0 that kpj = 0 for j > 1. Indeed, the embedding e1 : A1 → A is the kernel of p2 + · · · + pn . Consequently, the relations kp2 = · · · = kpn = 0 imply the relation k(p2 + · · · + pn ) = 0. Therefore, there exists a morphism l : Ker(1 − π) → A1 with le1 = k. Thus, l is a monomorphism and Ker(1 − π) ⊂ A1 . Therefore, (∗∗) A1 = Ker(1 − π) ⊕ (A1 ∩ Ker(π)) (Lemma 27.3 (3)) and A = Ker(1 − π) ⊕ (A1 ∩ Ker(π)) ⊕
n
Ai .
i=2
The projection A → Ker(1 − π) of this direct sum is equal to p1 e1 q, where qk = π and q is the projection A → Ker(1 − π) for the sum (∗) (Lemma 27.3 (1)). The projection A1 → Ker(1 − π) with respect to (∗∗) is e1 q by Lemma 27.3 (3). Verifying that, we obtain that ep1 e1 q : M → Ker(1 − π) and kpγ −1 : Ker(1 − π) → M are mutually inverse isomorphisms. However, ep1 e1 q is a composition of the embedding e from M in A with projection p1 e1 q from A onto Ker(1 − π). By Lemma 27.1 (1), we obtain A = M ⊕ (A1 ∩ Ker(π)) ⊕
n
Ai .
i=2
Therefore, M has the finite exchange property. Exercise 5 contains the assertion converse to Proposition 27.6. 3283
We say that an additive category E with infinite sums satisfies the weak Grothendieck condition if for every subscript set I, each nonzero object A, and every monomorphism A → Bi , there exist a finite i∈I
subset F of the set I and the commutative diagram C −−−−→
Bi
i∈F
⏐ ⏐
⏐ ⏐ Bi A −−−−→ i∈I
with nonzero morphism C → A.
Bi exists. Consequently, If E is a category with kernels, then by Lemma 27.3, the intersection A∩ i∈F Bi = 0. the existence of the indicated diagram is equivalent to the relation A ∩ i∈F
Proposition 27.7. Let E be an additive category with kernels that satisfies the weak Grothendieck condition. If an indecomposable object M has the finite exchange property, then M has the exchange property. Proof. Let A = M ⊕ N = Ai . For every subset J ⊆ I, we set A(J) = Ai . There exists a finite i∈I i∈J Ai = 0. By the finite exchange property, we have subset J in I such that M ∩ i∈J
A = M ⊕ N = A(J) ⊕ A(I \ J) = M ⊕
Ai
⊕ C,
i∈J
where Ai =
Ai
C
⊕ Bi , i ∈ J, A(I \ J) = C ⊕ and M⊕ Ai ⊕ C = Bi ⊕ C ⊕ Ai ⊕ C . i∈J
Now we have
i∈J
M∼ =
i∈J
Bi ⊕ C
i∈J
(Lemma 27.1). Since M is an indecomposable object, all objects Bi , i ∈ J, and C , except for one, are equal to zero. We assume that C = 0 and Bi = 0 for all i ∈ J. Since Ai = Ai for all i ∈ J, we have M = 0; this contradicts the choice of J. Consequently, C = 0, C = A(I \ J), and ∩ A(J) Ai ⊕ A(I \ J). If we assume that Ai = Ai for the subscripts i in I \ J, then A=M⊕ i∈J
A=M⊕
Ai ,
i∈I
where
Ai
⊂ Ai , i ∈ I. Therefore, M has the exchange property.
Now we can present the main result on the uniqueness decompositions. Theorem 27.8 (Walker–Warfield [290]). Let E be an additive category with kernels and infinite sums Ai , where the endomorphism ring of every and let E satisfy the weak Grothendieck condition. If M = i∈I
object Ai is local, then the following assertions hold. (a) Every indecomposable direct summand of the object M is isomorphic to one of the summands Ai . 3284
(b) If M =
j∈J
Bj and every summand Bj is indecomposable, then there exists a bijection θ : I → J
such that Ai ∼ = Bθ(i) for all i ∈ I. Proof. (a) Let M = A ⊕ B = Ai , where A is a nonzero indecomposable object. For a subset J ⊆ I, we i∈I denote by M (J) the object Ai . There exists a finite set J ⊂ I such that A ∩ M (J) = 0. It follows from i∈J
Proposition 27.6, Proposition 27.7, and Corollary 27.5 that the object M (J) has the exchange property. Consequently, M = M (J) ⊕ A ⊕ B , where A ⊂ A and B ⊂ B. Therefore, A = A ⊕ (A ∩ (M (J) ⊕ B ))
and A ∩ (M (J) ⊕ B ) = 0.
Since A is indecomposable, we have A = 0. Therefore, M = M (J) ⊕ B , B = B ⊕ (B ∩ M (J)), and M = A ⊕ B ⊕ (B ∩ M (J)). Therefore, M (J) ∼ = A ⊕ (B ∩ M (J)). By Theorem 27.2, the object B ∩ M (J) is equal to
m i=1
Bi , where Bi is an indecomposable object for
every i. It follows from Lemma 27.3 that End(A) and End(Bi ) have no nontrivial idempotents. It follows from Theorem 27.2 that A ∼ = Ai for some i. (b) By (a), every object Bj is isomorphic to some Ai . In particular, all Ai and B j have the exchange property. For subsets X ⊆ I and Y ⊆ J, we set M (X) = Ai and N (Y ) = Bj . Furthermore, i∈X
j∈Y
we set Ik = {i ∈ I | Ai ∼ = Ak } for k ∈ I. Then the sets Ik (k ∈ I) form a partition of the set I. Let Jk = {j ∈ J | Bj ∼ = Ak } for k ∈ I. Then {Jk | k ∈ I} is a partition of I. We take an arbitrary subscript k and assume that the set Ik is finite. In this case, the object M (Ik ) has the exchange property. Consequently, M = M (Ik ) ⊕ M (I \ Ik ) = M (Ik ) ⊕ N (J ) for some J ⊂ J (we take into account the indecomposability of Bj ). Now we have M (I \ Ik ) ∼ = N (J ) (Lemma 27.1). This implies Bj Ak , j ∈ J ; therefore j ∈ / Jk . Using the existence of the isomorphism M (Ik ) ∼ = N (J \ J ), we obtain Jk = J \ J . Therefore, M (Ik ) ∼ = N (Jk ), and it follows from Theorem 27.2 that |Ik | = |Jk |. Now we assume that Ik is an infinite set. Denote by Δk the set {j ∈ J | M = Ak ⊕ N (J \ {j})}. There exists a finite set F ⊂ J with Ak ∩ N (F ) = 0 (this follows from the use of the weak Grothendieck condition). If j ∈ / F , then the inclusion F ⊂ J \{j} implies the relation Ak ∩N (J \{j}) = 0. Consequently, Δk ⊂ F ; therefore, every Δk is finite. Now let t ∈ Jk . Then Bt ∩ M (G) = 0 for some finite subset G in I. In addition, it follows from the exchange property for M (G) that M = M (G) ⊕ N (J \ H), where H is some subset in J. Since M (G) ∼ = N (H), it follows from Theorem 27.2 that H is finite. It is clear that t is contained in H. Since N (H \ {t}) has the exchange property, we have M = N (H \ {t}) ⊕ Bt ⊕ N (J \ H) = M (G) ⊕ N (J \ H) = N (H \ {t}) ⊕ Ag ⊕ N (J \ H) for some g ∈ G (Lemma 27.4). Therefore, t ∈ Δg . In addition, Ag ∼ = Bt ∼ = Ak ; therefore g ∈ Ik . We have proved the relation Jk = Δg . Since Ik is infinite and every Δg is finite and nonempty, we have g∈Ik
|Jk | ≤ |I. Using a similar argument, we obtain |Jk | = |Ik |. Before, we introduced several not so well known notions and agreed with the following terminology. Let f : A → B and g : X → B be two morphisms. If there exists a morphism h : X → A with g = hf , then we say that g factors through f or A. A small object of the category E is an object S such that every morphism S → Ai can be factored i∈I through the embedding Ai → Ai for some finite subset F of the set I. i∈F
i∈I
The following definition transfers the module property from Exercise 1 to categories. 3285
An object M is said to be finitely approximable if every monomorphism f : L → M is an isomorphism iff for every small object S, every morphism S → M can be factored through f . It is clear that any small object is a finitely approximable object. Lemma 27.9. Let E be an additive category with kernels. If M is a nonzero finitely approximable object Ai , then there exists a finite subset F in I such that M ∩ Ai = and M ⊂ 0. i∈I
Proof. Let e be a monomorphism from M into
i∈I
i∈F
Ai . Since the monomorphism 0 → M is not an
isomorphism, there exist a small object S and a nonzero morphism f : S → M . The morphism f e can Ai and f induces the nonzero morphism S → M ∩ Ai . be factored through some finite sum i∈F
i∈F
Consequently, the intersection is not equal to zero. Now we can assert that Proposition 27.7 and Theorem 27.8 remain valid if we replace the weak Grothendieck condition by the property that the object M is finitely approximable or all objects Ai (i ∈ I) are finitely approximable (for more details, see the paper of Walker and Warfield [290]). To formulate one more main result of this section, we define the following notion. An object M is said to be countably approximable if there exist a countable family of small objects Si and morphisms gi : Si → M (i ≥ 1) such that every monomorphism f : L → M is an isomorphism iff every gi can be factored through f . The proof of the following theorem of Walker and Warfield is very difficult. We recommend that the reader see it in the original paper of Walker and Warfield [290]. Theorem 27.10. Let E be an additive category with kernels and infinite sums, let E satisfy the weak Grothendieck condition, and let M = Ai , where for every i, the object Ai is countably approximable i∈I
and the ring End(Ai ) is local. Then every direct summand of the object M is isomorphic to the direct sum Ai for some subset J ⊆ I. i∈J
Consequently, any two direct decompositions of the object M have isomorphic refinements. As usual, the Krull–Schmidt theorem means the assertion on an isomorphism of two direct decompositions of a module or an object of the category with indecomposable summands. In this case, we say that the module or the object satisfies the Krull–Schmidt theorem. The paper of Walker and Warfield [290] contains a detailed review of the studies related to this theorem. We present the generalizations of two known theorems of Krull–Schmidt type. The first theorem is the following result of Azumaya [23]. Assume that a module M has a decomposition Ai (I is a finite or infinite set) such that the i∈I
endomorphism ring of every Ai is local. Then any another indecomposable summand of the module M is isomorphic to Ai for some i ∈ I; in addition, if M is the direct sum of indecomposable submodules Bj , j ∈ J, then there exists a bijective mapping f : I → J such that Ai ∼ = Bf (i) for all i ∈ I. The second theorem is due to Crawley, J´onsson, and Warfield; the theorem can be formulated as follows (see the work of Anderson and Fuller [1, Theorem 26.5]). If the conditions of the previous theorem hold and each of the modules Ai is countably generated, then any two direct decompositions of the module M have isomorphic refinements. In particular, if N is a direct summand of the module M , then there exists a subset J ⊆ I such that N is isomorphic to the direct sum of modules Ai , i ∈ J. Crawley and J´ onsson [52] have published some theorems on the uniqueness of decompositions and isomorphisms of refinements for arbitrary algebraic systems. The main condition of their paper is the exchange property. This property and the so-called cancellation property and the substitution property are studied in [69, 295, 297, 298, 304] and other papers (for more details, see the references in these papers). Different variants of the Krull–Schmidt theorem for categories were obtained by Atiyah [21], Gabriel [94], Bass [31], Warfield [296], Arnold–Hunter–Richman [15], and Arnold [7]. 3286
The open question on the uniqueness of direct decompositions is twofold: (1) Whether two given decompositions are isomorphic to one another. (2) Whether a direct summand of a given direct decomposition is a direct sum of the corresponding summands. Exercise 1. (a) Any finitely generated module over any ring is a small module. In general, the converse is not true. Every module is the least upper bound of submodules that are small modules. (b) A small module over a discrete valuation domain is a finitely generated module. Exercise 2. Transfer property (b) from Sec. 2 (Part I) and Proposition 2.4 (Part I) to additive categories. Exercise 3. Any direct summand of an object with the exchange property has the exchange property. Ai , then every Ai is finitely approximable. Exercise 4. If M is a finitely approximable object and M = i∈I
If the category E has kernels, then the converse assertion also holds. Exercise 5 (Warfield [295], Walker–Warfield [290]). An indecomposable finitely approximable object of an additive category with kernels has the exchange property iff the endomorphism ring of the object is local. 28.
Isotype, Nice, and Balanced Submodules
We define and study submodules of some special types. They are very useful for the theory, which will be developed later. First, we specify the notion of the height of an element and consider some related facts. We call a sequence {σi }i≥0 a height sequence provided every σi is either an ordinal number or the symbol ∞ and, in addition, (1) if σi = ∞, then σi+1 = ∞; (2) if σi+1 = ∞, then σi < σi+1 . The elements σi are called the coordinates of the given height sequence. We can define a natural partial order on the set of all height sequences. Let u = {σi } and v = {τi } be two height sequences. We assume that u ≤ v if σi ≤ τi for all i ≥ 0. We also assume that σ < ∞ for every ordinal number σ. In fact, we obtain a lattice where the greatest lower bound inf(u, v) is equal to the sequence (ρ0 , ρ1 , . . .) with ρi = min(σi , τi ). Similarly, the least upper bound sup(u, v) corresponds to the pointwise maximum. If σi + 1 < σi+1 , then we say that the sequence u has a jump between σi and σi+1 . Two height sequences u and v are said to be equivalent (the notation is u ∼ v) if there exist two positive integers n and m such that σn+i = τm+i for i = 0, 1, 2, . . .. This yields an equivalence relation on the set of all height sequences. We denote by pn u (n ≥ 0) the height sequence (σn , σn+1 , . . .). It is clear that u ∼ pn u. Every element of the module M is related to some height sequence. First, we define the submodule pσ M for every ordinal number σ. For any nonnegative integer n, the submodule pn M was introduced in Sec. 4 (Part I) as the set {pn x | x ∈ M }. Now we set pσ M = p(pσ−1 M ) provided σ − 1 exists, and pσ M = pτ M for any limit ordinal number σ. Using cardinalities, we obtain that there exists the least τ 0. If M is a primary module, then X satisfies the minimum condition. We continue the direction considered in Theorem 30.3. Precisely, we show that a certain partition of the set X induces a direct decomposition of the module M . For an element y ∈ X, we set Xy = {x ∈ X | y ≤ x}. (a) If M is a simply presented primary module and L is the set of minimal elements in X, then M= RXy . y∈L
For distinct y, z ∈ L, the sets Xy and Xz are disjoint. By the minimum condition, they cover the set X. Therefore, the sets Xy (y ∈ L) form a partition of the set X. The equivalence relation on X defined earlier can be specified as follows. We say that two elements x, y ∈ X are equivalent if pm x = pn y = 0 for some m, n > 0. (This specification refers to only elements of finite order in X.) Then the sets Xy coincide with equivalence classes for the defined new equivalence relation on X. Similarly to Theorem 30.3, we obtain that M is a direct sum of modules RXy , y ∈ L. (b) If M is a simply presented module with generator system X and Y is some subset in X, then RY and M/RY are simply presented modules. Indeed, the module RY is generated by the set Y , and every relation between elements of Y has the form px = 0 or px = y. The factor module M/RY can be defined by generators x ¯ = x + Ry, x ∈ X, by relations of the form p¯ x = 0 and p¯ x = y¯, and by additional relations of the form x ¯ = 0. (c) Every nonzero element a of a simply presented module M with generator system X can be uniquely represented in the form a = v1 x1 + · · · + vk xk , where x1 , . . . , xk are distinct elements of X and v1 , . . . , vk are invertible elements of the ring R. First, we note that if z1 , . . . , zn are some distinct elements in X and, for example, z1 is a maximal element in this set with respect to the order indicated before (a), then z1 ∈ / Rz2 + · · · + Rzn . By properties (1)–(3) of the set X, the existence of such a representation is obvious. We assume that there exists another representation of the element a. By convention, it can be presented in the form a = r1 x1 + · · · + rk xk + rk+1 xk+1 + · · · + rl xl , where either ri = 0 or ri is an invertible element. It follows from the property of maximal elements just presented that each of the elements xk+1 , . . . , xl cannot be maximal in the set x1 , . . . , xl . Consequently, we can assume that x1 is maximal. Then x1 ∈ / Rx2 + · · · + Rxl . There exists a homomorphism ϕ from the module Rx1 + · · · + Rxl into the quasicyclic module R(p∞ ) that maps x1 into an element of order p and maps x2 , . . . , xl into 0. Since R(p∞ ) is an injective module, we can assume that ϕ is defined on M . We have ϕa = v1 (ϕx1 ) = r1 (ϕx1 ), which implies r1 = v1 . The induction on k completes the proof. (d) If we consider a unique representation from (c) and a ∈ pσ M , then xi ∈ pσ M for all i = 1, . . . , k. For σ = 0, the assertion is trivial. We assume that the assertion holds for all ordinal numbers less than σ. The case of the limit ordinal number σ is easy. Assume that σ − 1 exists and a = pb for some b = u1 y1 + · · · + ul yl ∈ pσ−1 M , where y1 , . . . , yl are distinct elements from X and u1 , . . . , ul are invertible elements from R. By the induction hypothesis, yj ∈ pσ−1 M for j = 1, . . . , l. The element a can be represented in the form w1 z1 + · · · + wm zm with distinct elements z1 , . . . , zm of X and invertible elements w1 , . . . , wm . Each of the elements z1 , . . . , zm has the form pn yj for some j and n ≥ 1, whence z1 , . . . , zm ∈ pσ M . It follows from (c) that these elements z are equal to the elements x, i.e., x1 , . . . , xk ∈ pσ M . (e) Let M be a reduced simply presented primary module such that the length of M is a limit ordinal number. Then M is a direct sum of simply presented modules whose lengths are less than the length of M . The assertion remains valid if we replace the length by the Ulm length (the notions of the lengths were introduced in Sec. 28). The required decomposition follows from Part (a). It follows from (d) that the last nonzero Ulm submodule of the module RAy is generated by the element y. Consequently, the lengths of the modules RAy are unlimit ordinal numbers. Consequently, they are less than the length of the module M . 3304
(f) If M is a simply presented module, then for every ordinal number σ, the modules pσ M and M/pσ M are also simply presented. In addition, if the module M is a reduced primary module, then all Ulm factors of M are direct sums of cyclic modules. Let X be a generator system modules M . We set Y = {y ∈ X | h(y) ≥ σ}. pσ M
= RY and the result follows from (b). Since M σ = pωσ M for every σ, Ulm It follows from (d) that submodules and Ulm factors of the module M are simply presented. Ulm factors of a reduced primary module have no elements of infinite height. It remains to prove that the simply presented primary module M without elements of infinite height is a direct sum of cyclic modules. Indeed, since the length of the module M does not exceed ω, we have that, by (e), M is the direct sum of bounded modules; therefore M is a direct sum of cyclic modules. (g) If Y is some subset of a generator system X of a simply presented module M , then RY is a nice submodule in M . Similarly to (c), the element a ∈ M \ A can be represented in the form a = u1 x1 + · · · + uk xk + v1 y1 + · · · + vl yl , where xi and yj are elements of X \ Y and Y , respectively, and ui and vj are invertible elements of the ring R. For simplicity, we assume that x1 , . . . , xs (s ≤ k) are not contained in A and all remaining elements x are contained in A. We assert that the element b = u1 x1 + · · · + us xs of the residue class a + A is proper with respect to A. Indeed, we take an arbitrary element c of A and represent the sum us+1 xs+1 + · · · + uk xk + v1 y1 + · · · + vl yl + c in the form of the sum from Part (c): w1 z1 + · · · + wm zm . Then u1 x1 + · · · + us xs + w1 z1 + · · · + wm zm is a unique expression for the element a + c indicated in (c). It follows from Part (d) that h(a + c) = min{h(xi ), h(zj ) | i = 1, . . . , s; j = 1, . . . , m} ≤ min{h(xi ) | i = 1, . . . , s} = h(b). Reduced, simply presented primary modules coincide with totally projective modules. A reduced primary module M is said to be totally projective if pσ Ext(M/pσ M, N ) = 0 for every ordinal number σ and every module N . These notions first appeared in the theory of Abelian p p -modules; see Sec. 4 (Part I)). groups, (equivalently, primary groups, or primary Zp -modules, or primary Z With corresponding exchanges, the theory of p-groups can be extended to primary modules over an arbitrary discrete valuation domain. In a more general context, this was discussed in Sec. 4 (Part I) and in the remarks to the present chapter. We can better understand the role and importance of the results of the central Section, Sec. 32, of this chapter if we cover several aspects of the theory of p-groups related to the famous Ulm theorem. In 1933, Ulm [282] has proved that two countable reduced p-groups A and B are isomorphic to one another iff the corresponding Ulm–Kaplansky invariants of the groups A and B coincide. Ten years before, Pr¨ ufer [244] proved that Ulm factors of countable p-groups are direct sums of cyclic groups. The theorems of Ulm and Pr¨ ufer complement the Zippin theorem [317] on the existence of a countable p-group with a given Ulm sequence. These three theorems form a complete structural theory of countable p-groups. We can say that they give a complete classification of countable p-groups. The problem on the existence of p-groups with an arbitrary given Ulm sequence was solved by Kulikov [181] and Fuchs [87] (see [90, Theorem 76.1]). As for the classification of (noncountable) p-groups, Kolettis made the first serious step toward this only in 1960 [166]. He extended the Ulm–Zippin theory to direct sums of countable p-groups. These groups are also distinguished by their Ulm–Kaplansky invariants. It is natural to formulate the following problem: describe (sufficiently large) classes of p-groups with similar properties. In the process of homological consideration, Nunke ([238, 239]) defined totally projective groups. He characterized direct sums of countable p-groups as totally projective groups A such that pω1 A = 0, where ω1 is the first uncountable ordinal number. Later, Hill [137] extended Ulm’s theorem to totally projective groups (see also Walker [291]). Crawley and Hales ([50, 51]) used another approach and discovered simply presented p-groups (“T -groups” in their terminology); they proved Ulm’s theorem 3305
for such groups. Using the result of Hill, they proved that these groups coincide with totally projective groups in the reduced case. For this reason, we consider the words “reduced simply presented primary module” as idential with “totally projective module.” Totally projective modules can be characterized in terms of the existence of certain families of nice submodules. We present necessary definitions. We say that a primary module M satisfies the Hill condition (or “has a nice system,” or “satisfies the third countability axiom”) if M admits a system N of nice submodules such that (a) 0 ∈ N ; (b) if {Ai | i ∈ I} is an arbitrary set from N , then Ai ∈ N ; i∈I
(c) for every submodule A ∈ N and any countable or finite subset X of the module M , there exists a submodule C ∈ N such that A, X ⊆ C and C/A is a countably generated module. (The definition of a countably generated module is presented in Sec. 8 of Part I.) We give one more definition. Let M be a primary module, and let 0 = A0 ⊂ A1 ⊂ · · · ⊂ Aσ ⊂ · · · ⊂ Aμ = M be a well-ordered, properly ascending chain of nice submodules of the module M satisfying the following conditions: (a) Aσ+1 /A σ is a cyclic module of order p for every σ < μ; Aτ if σ is a limit ordinal number. (b) Aσ = τ h(pn xj )
for all n. Then the subordinate {ps(j) xj | j = 1, 2, . . .} is nice (see Exercise 2). In connection with the following proposition, let us recall that totally projective modules are defined before Theorem 30.4. We also note that if A is a nice submodule in M , then M/A is a reduced module (we assume that M is a reduced module). Proposition 31.5 (Hunter–Richman [150]). Let X and Y be two decomposition bases with X ⊂ RY . Then there exists a subordinate Z in X such that RZ is a nice submodule in RY and RY /RZ is a totally projective module. Proof. Using Lemma 31.4 and the property that a countably generated, reduced primary module is a totally projective module, we can assume that Y is a noncountable set. Let us completely order the set Y , i.e., let Y = {yσ | σ < λ}, where λ is the least ordinal number of cardinality |Y |. For every σ < λ, we define a set Yσ by taking the closure of the set Yσ ∪ {yσ } in the sense of Lemma 31.3 as Yσ+1 and by setting Yσ = Yτ for a limit ordinal number σ. Now we set Xσ = X ∩ RYσ . It follows from Lemma 31.3 τ τ . There is an extension of ϕ such that x is mapped onto w + z. We set A1 = A + Rx and B1 = B + R(w + z). The submodules A1 and B1 are nice, since A and B are nice, and A1 /A and B1 /B are finitely generated by Lemma 28.3 and Proposition 28.4. If σ < τ , then x ∈ pσ+1 M and A1 (σ) = A(σ). Similarly, we obtain B1 (σ) = B(σ). If σ > τ , then the same relations follow from the property that the element x is proper with respect to A and w + z is proper with respect to B. If σ = τ , then x − y ∈ A1 (σ) and 0 = tσ (x − y + A(σ)) = z + B(σ) ∈ B1 (σ)/B(σ). The modules A1 /A and B1 /B are cyclic modules of order p. Since there exist the canonical epimorphism A1 /A → A1 (σ)/A(σ) and a similar epimorphism for B, it follows that A1 (σ)/A(σ) and B1 (σ)/B(σ) are also cyclic of order p. Consequently, tσ induces an isomorphism, which is what was required. This proves (1), (2) and (3). The proof of (4) follows from (3). Case 2. h(px) = τ + 1. Let ϕ(px) = pw with h(w) ≥ τ . Then h(w) = τ , since h(pw) = τ + 1. If h(w + b) ≥ τ + 1 for some b ∈ B, then h(b) = τ and b = ϕ(a), where h(a) = τ and h(x + a) = τ . Therefore, x + a is proper with respect to A, h(p(x + a)) > τ + 1, and we obtain Case 1. Consequently, w is proper with respect to B, and it is obvious that w ∈ / B. There is an extension of ϕ which maps x into w. We set A1 = A + Rx and B1 = B + Rw. If σ = τ , then, similarly to (1), we obtain A1 (σ) = A(σ) and B1 (σ) = B(σ). If σ = τ , then the relation p(m + c + x) = 0 does not hold for any elements c ∈ A and m ∈ pσ+1 M . Once again we have A1 (σ) = A(σ) and B1 (σ) = B(σ), which completes the proof. Let A and B be submodules of the modules M and N , respectively, and let ϕ : A → B be some isomorphism. We assume that ϕ extends to an isomorphism ψ : M → N . Since pσ M and (pσ M )[p] are completely invariant submodules, ψ induces the isomorphisms (pσ M )[p] → (pσ N )[p] and A(σ) → B(σ). Therefore, we have fσ (M, A) = fσ (N, B) for relative invariants. The following theorem was originally proved by Hill [137]; he additionally assumed that M and N are primary modules. The presented formulation and proof are due to Walker [291]. Theorem 31.9. Let M/A and N/B be two totally projective modules, where A and B are nice submodules and fσ (M, A) = fσ (N, B) for all σ. Then every height-preserving isomorphism ϕ : A → B extends to an isomorphism M → N . 3312
Proof. Let
tσ : (pσ M )[p]/A(σ) → (pσ N )[p]/B(σ) be an isomorphism for every ordinal number σ. The modules M/A and N/B satisfy the Hill condition (Theorem 30.4). Let E and D be two systems of nice submodules for the modules M/A and N/B from the definition of the Hill condition before Theorem 30.4. We consider the family F of all height-preserving isomorphisms C → D such that (a) C/A ∈ E and D/B ∈ D; (b) for every σ, tσ induces an isomorphism C(σ)/A(σ) → D(σ)/B(σ). On the set of such isomorphisms, we define a partial order in the standard way. Precisely, if ϕi : Ci → Di are isomorphisms (i = 1, 2), then ϕ1 ≤ ϕ2 ⇐⇒ C1 ⊆ C2 and the restriction of ϕ2 to C1 coincides with ϕ1 . It follows from the Zorn lemma that there exists the maximal such isomorphism ϕ0 : C → D. Condition (b) implies the relation fσ (M, C) = fσ (N, D) for all σ. The submodules C and D are nice by Lemma 28.3. For every σ, tσ induces the isomorphism (pσ M )[p]/C(σ) → (pσ N )[p]/D(σ). We assume that M = C. Let z ∈ M \ C. By Lemma 31.8, there exist nice submodules A1 = C + Rz, B1 and a height-preserving isomorphism A1 → B1 such that the isomorphism extends ϕ0 and tσ induces the isomorphism A1 (σ)/C(σ) → B1 (σ)/D(σ). Consequently, the isomorphism A1 → B1 induces the isomorphism A1 (σ)/A(σ) → B1 (σ)/B(σ). We have the system E and nice submodules A and A1 . Consequently, it follows from Lemma 28.3 ∞ Rx1i . There that there exists a nice submodule C1 such that A1 ⊂ C1 , C1 /A ∈ E, and C1 = A1 + i=1
exists a height-preserving isomorphism A1 + Rx11 → B2 that satisfies condition (b) and extends the isomorphism A1 → B1 . In a similar way, we use the system D and nice submodules B and B2 in order ∞ Ry2i and D2 /B ∈ D. There exists a to obtain a nice submodule D2 with the properties D2 = B2 + i=1
height-preserving isomorphism A2 → B2 + Ry21 that extends the previous isomorphism and satisfies (b). ∞ Furthermore, we have C2 = A2 + Rx2i with C2 /A ∈ E. Similarly, there exists a height-preserving i=1
isomorphism A2 + Rx12 + Rx21 → B3 that extends the previous isomorphism and satisfies (b). Now ∞ we have D3 = B3 + Ry3i with D3 /B ∈ D. Thus, there exists a height-preserving isomorphism i=1
A3 → B3 + Ry22 + Ry31 that extends the previous and satisfies (b). Proceeding in such a isomorphism way, we obtain a height-preserving isomorphism C = A → Di that extends ϕ0 and satisfies (b). i i However, ( Ci )/A ∈ E and ( Di )/B ∈ D. Therefore, C = M and D = N , which completes the proof. Corollary 31.10. If A is a nice submodule of the module M such that M/A is a totally projective module, then every automorphism of the module A that preserves heights in M extends to an automorphism of the module M . Corollary 31.11. Let M and N be two modules, A be a nice submodule in M , and M/A be a totally projective module. If ϕ : A → N is a homomorphism that does not decrease heights, then ϕ can be extended to a homomorphism from M into N . Proof. We extend ϕ to a homomorphism ψ : A ⊕ N → A ⊕ N setting ψ(a, x) = (a, ϕ(a) + x). The homomorphism ψ is a height-preserving in M ⊕ N automorphism of the submodule A ⊕ N . By the previous corollary, ψ extends to some automorphism α of the module M ⊕ N . Let θ : M → M ⊕ N be 3313
the corresponding embedding and let π : M ⊕ N → N be the corresponding projection. Then θαπ is the required extension of ϕ. Exercise 1. Prove that every subordinate of a nice decomposition basis is nice. Exercise 2 (Hunter–Richman–Walker [151]). We assume that a decomposition basis X of the module M satisfies the following condition. For a given ordinal number σ, there exist no more than finitely many ordinal numbers τ < σ such that h(x) = τ and h(px) ≥ σ for some x ∈ X. Prove that RX is a nice submodule in M . Exercise 3. Let X be a decomposition basis of the module M . Verify that dimRp (Iσ (RX)) coincides with the number of elements in X such that the Ulm sequences of these elements have a jump at σ in the sense indicated before Lemma 31.7. The following three exercises are taken from the paper of Hunter and Richman [150]. Exercise 4. Let M be a module with nice decomposition basis. Then every decomposition basis of the module M has a nice subordinate. Exercise 5. Let Z be a finite subset of some decomposition basis. Then RZ is a nice submodule. Exercise 6. Any two finite decomposition bases contain subordinates that generate equal submodules. Exercise 7 (Files [70]). If M is an exact module of rank > 1, then M has no nice free submodule. 32.
Warfield Modules
We define Warfield modules and prove that they have complete systems of invariants consisting of cardinal numbers. The independence of this system is considered in remarks at the end of this chapter. As earlier, we assume that M is a reduced mixed module (unless otherwise specified). First, we formulate the following result on decomposition bases. Proposition 32.1. For a module M , let Y be a nice decomposition basis of M and let X be a decomposition basis of M . Then there exists a nice subordinate Z in X with RZ ⊆ RY . In addition, if M/RY is a totally projective module, then there exists a nice subordinate Z in X such that RZ ⊆ RY and M/RZ is a totally projective module. Proof. Taking the subordinate, if necessary, we can assume that X ⊂ RY . By Proposition 31.5, there exists a subordinate Z in X such that RZ is a nice submodule in RY and RY /RZ is a totally projective module. Since RY is a nice submodule in M , RZ is also a nice submodule in M , i.e., Z is a nice subordinate. Let M/RY be a totally projective module. The module M/RZ can be considered as an extension of the nice submodule RY /RZ by the module M/RY . By Corollary 30.5, M/RZ is a totally projective module (see also Lemma 28.3). A mixed module M is called a Warfield module if M is isomorphic to a direct summand of some simply presented module. Every simply presented module is a Warfield module. There exist simply presented modules of rank 1 that are not Warfield modules (an example is presented in [305]). Theorem 32.2 (Warfield [305]). (1) A Warfield module M contains a nice decomposition basis X such that every subordinate X in X is nice and M/RX is a totally projective module. (2) If a module M contains a nice decomposition basis X such that M/RX is a totally projective module, then M is a Warfield module. Proof. (1) There exist a module N and a simply presented module L such that M ⊕ N = L. By Corollary 31.2, M and N have decomposition bases W and Z, respectively. We set Y = W ∪ Z. Let V be a decomposition basis for L such that it is a subset of some generator system of the module L that satisfies conditions (1)–(3) before Lemma 30.1. We can always do this as follows. By Theorem 30.3, the 3314
module L is a direct sum of simply presented modules Li of rank 1, i ∈ I. In every Li , we choose one element xi of infinite order from some generator system of the modules Li . We set V = {xi | i ∈ I}. Since every subordinate in V is a subset of the same generator system, it follows from Lemma 31.7 that we can assume that V is a lower decomposition basis. By properties (g) and (b) from Sec. 30, V is a nice basis and L/RV is a totally projective module (see Theorem 30.4, the text before the theorem, and the remark before Proposition 31.5). We can choose a subordinate Y in Y that satisfies the following properties: (1) Y ⊆ RV ; (2) Y is a lower decomposition basis; (3) there exists a bijection t : Y → V such that U (y) and U (t(y)) are equivalent and U (y) ≥ U (t(y)) for every y ∈ Y . Using Proposition 32.1 and passing to further subordinates (which does not change conditions (1)–(3)), we can assume that Y is a nice subordinate and L/RY is a totally projective module. We can also replace V by some subordinate V such that U (y) = U (pn t(y)) provided pn t(y) ∈ V . In this case, we obtain a height-preserving isomorphism ϕ : RV → RY (it extends the bijection inverse to t). This implies the following isomorphism of Rp -spaces: Iσ (RV ) ∼ = Iσ (RY ) for all ordinal numbers σ; this follows from the definition of the monomorphism κσ before Lemma 31.6. Consequently, if these spaces are finite-dimensional, then fσ (L, RV ) = fσ (L, RY ) by relation (4) from Sec. 31. If they are infinite-dimensional, then we take into account the property that V and Y are lower decomposition bases and obtain fσ (L, RV ) = f (σ, L) = fσ (L, RY ). The conditions of Theorem 31.9 hold. Therefore, ϕ extends to an automorphism of the module L. In addition, Y turns out to be a subset of some generator system modules L satisfying conditions (1)–(3) before Lemma 30.1. We set X = Y ∩ M . Since Y is a subordinate for Y , X is a subordinate for W . Consequently, X is a decomposition basis for M . Since X is a subset of some generator system of L, every subordinate X of X is a subset of the same generator system for L. Consequently, RX is a nice submodule in M according to property (g) from Sec. 30. Let Z = Y ∩ N . Then L/RY = L/R(X ∪ Z ) = L/(RX ⊕ RZ ) ∼ = M/RX ⊕ N/RZ . Consequently, M/RX is a totally projective module as a direct summand of a totally projective module. This proves that the decomposition basis X of the module M has all the properties required in (1). (2) Considering Proposition 31.1 and the remark after it, we can take a simply presented module N with nice decomposition basis Y and the totally projective factor module N/RY (see the beginning of the proof of (1)). In addition, there exists a height-preserving isomorphism between RX and RY . (Therefore M and N are isomorphic to one another in Warf.) Now we note that the class of totally projective modules is closed with respect to direct sums. Consequently, it follows from Theorem 30.2 that there exist totally projective modules with arbtrarily large Ulm–Kaplansky invariants. More precisely, let T be a totally projective module such that f (σ, T ) = 0 provided f (σ, M ) = f (σ, N ) = 0 for every σ. Otherwise, let f (σ, T ) be an infinite cardinal number with f (σ, T ) > min(f (σ, M ), f (σ, N )). We consider the modules M ⊕ T and N ⊕ T . They have the same Ulm–Kaplansky invariants. As for the Ulm–Kaplansky invariants of the module M ⊕ T with respect to RX, it follows from relations (2) and (4) in Sec. 31 that the invariants are equal to the Ulm–Kaplansky invariants of the module M ⊕ T . The same is true for N ⊕ T and RY . Therefore, we can apply Theorem 31.9 to the modules M ⊕ T and N ⊕ T 3315
and their submodules RX and RY . As a result, we obtain M ⊕ T ∼ = N ⊕ T , where N ⊕ T is a simply presented module. In fact, Part (a) of the following corollary was proved in Part (2) of Theorem 32.2; Part (b) follows from Proposition 32.1 and Part (1) of this theorem. Corollary 32.3. (a) For every Warfield module M , there exists a totally projective module T such that M ⊕ T is a simply presented module. (b) Every decomposition basis X of the Warfield module M contains a nice subordinate Z such that M/RZ is a totally projective module. In connection with (a), we mention a deep result of Hunter, Richman, and Walker [152], which states that every Warfield module is a direct sum of a simply presented module and a Warfield module of at most countable rank. The following two corollaries are related to category-theoretical properties of Warfield modules. Corollary 32.4. If M is a Warfield module and N is an arbitrary module, then we have the following equality of morphism groups: HomW (M, N ) = HomW (M, N ). Proof. We always have the inclusion HomW (M, N ) ⊆ HomW (M, N ); the meaning of the inclusion is explained before Corollary 29.7. We take a decomposition basis X of the module M , which exists by Part 1) of Theorem 32.2. We assume that ϕ¯ ∈ HomW (M, N ) and a homomorphism ϕ1 : A → N represents ϕ. ¯ This means that A is a subtorsion-free module in M such that M/A is a primary module and ϕ1 does not decrease the heights of elements. For every x ∈ X, there exists an integer n ≥ 0 (depending on x) with pn x ∈ A. Denote by X the set of such elements pn x for all x. Let ϕ be the restriction of ϕ1 to X . Then RX is a nice submodule and M/RX is a totally projective module. By Corollary 31.11, ϕ extends to a homomorphism ψ : M → N . We obtain ψ¯ = ϕ. ¯ Therefore, ϕ¯ ∈ HomW (M, N ). The Warfield module M is a direct summand of a simply presented module; therefore, it is a direct summand of a completely decomposable module. Consequently, M has Warfield invariants g(e, M ) (see Secs. 29–31 on these invariants). Corollary 32.5. For two Warfield modules M and N , the following conditions are equivalent: (1) g(e, M ) = g(e, N ) for every equivalence class e of height sequences; (2) M ∼ = N in the category Warf ; (3) M ∼ = N in the category Walk ; (4) there exist primary modules S and T such that M ⊕ T ∼ = N ⊕ S; (5) there exists a totally projective module T such that M ⊕ T ∼ = N ⊕ T. Proof. Conditions (1) and (2) are equivalent by Corollary 29.10. Conditions (2) and (3) are equivalent by Corollary 32.4. Conditions (3) and (4) are equivalent by Proposition 29.2. The implication (1) =⇒ (5) follows from the proof of Part (2) of Theorem 32.2. In addition, we also need to consider the paragraph after the proof of Corollary 31.2. Finally, the implication (5) =⇒ (4) is trivial. If ϕ is an isomorphism between some modules M and N , then the restriction of ϕ is an isomorphism from (pσ M )[p] onto (pσ N )[p]. Consequently, ϕ induces an isomorphism of the corresponding factor modules; this implies the equality of Ulm–Kaplansky invariants f (σ, M ) = f (σ, N ) for every σ. It follows from Corollary 32.5 that the invariants g(e, M ) of Warfield modules satisfy an analogous property. It is remarkable that it suffices to consider the cardinal numbers f (σ, M ) and g(e, M ) in order to distinguish Warfield modules. We are ready to present the main classification theorem of this chapter. 3316
Theorem 32.6 (Warfield [299, 303, 305]). Two Warfield modules M and N are isomorphic to one another iff f (σ, M ) = f (σ, N ) for every ordinal number σ and g(e, M ) = g(e, N ) for every equivalence class e of height sequences. Proof. It suffices to prove the sufficiency. Let X and Y be the decomposition bases of the modules M and N , respectively, considered in Part (1) of Theorem 32.2. Passing to subordinates, if necessary, we can assume that X and Y are lower bases by Lemma 31.7. Since the invariants g(e, M ) coincide with g(e, N ), it follows from remarks after the proof of Corollary 31.2 that there exists a bijection t : X → Y such that U (x) = U (t(x)) for all x ∈ X. The bijection t naturally extends to a height-preserving isomorphism ϕ : RX → RY . Then we use an argument similar to the argument used in the proof of Part (1) of Theorem 32.2. Precisely, we have the isomorphism of Rp -spaces Iσ (RX) ∼ = Iσ (RY ) for all ordinal numbers σ. If these spaces are finite-dimensional, then fσ (M, RX) = fσ (N, RY ). For infinite dimensions, we have fσ (M, RX) = f (σ, M ) = f (σ, N ) = fσ (N, RY ) (X and Y are lower bases!). By Theorem 31.9, ϕ extends to an isomorphism from M onto N . We can say that two Warfield modules are isomorphic to one another iff their Ulm–Kaplansky invariants coincide and their Warfield invariants coincide. Let us present other formulations of Theorem 32.6. Corollary 32.7. For two Warfield modules M and N , the following conditions are equivalent: (1) M ∼ = N; (2) M and N have equal Ulm–Kaplansky invariants and are isomorphic to one another in Warf ; (3) t(M ) ∼ = t(N ) and Warfield invariants of modules M and N coincide. Proof. The equivalence of (1) and (2) follows from Corollary 32.5 and Theorem 32.6. The equivalence of (1) and (3) is contained in Theorem 32.6 if we consider that Ulm–Kaplansky invariants of every module are equal to the similar invariants of the torsion submodule of the module. We apply the obtained results to modules of rank 1; then we apply them to countably generated modules, which were already considered before Theorem 8.2 (Part I). Corollary 32.8. (a) Assume that M is a Warfield module of rank 1, x is an element of infinite order from M , and a homomorphism ϕ : Rx → N does not decrease heights. Then ϕ extends to a homomorphism M → N. (b) A module M of rank 1 is a Warfield module iff there exists an element x in M of infinite order such that M/Rx is a totally projective module. (c) Two Warfield modules M and N of rank 1 are isomorphic to one another iff they have the same Ulm–Kaplansky invariants and U (M ) = U (N ). Proof. (a) By Proposition 28.4, the submodule Rx is nice in M . Now we use Corollary 31.11. Property (b) follows from Proposition 28.4 and Theorem 32.2. As for (c), it follows from Proposition 29.6 that M ∼ =N in Warf ⇐⇒ U (M ) = U (N ). Therefore, the result follows from Corollary 32.7. Let M be a module of rank 1 such that t(M ) is a totally projective module. If x is an element of infinite order, then M/Rx is a totally projective module, since it is a countably generated extension of a totally projective module. Consequently, M is a Warfield module and Corollary 32.8 applies to M (see paper Wallace’s [293]). Corollary 32.9. A countably generated module M is a Warfield module iff M has a decomposition basis. Proof. Any Warfield module always has a decomposition basis. Conversely, if M has a decomposition basis X, then X is a countable or finite set. It follows from Lemma 31.4 that X has a nice subordinate X . Since M/RX is a reduced, countably generated module, it is a totally projective module (this was mentioned at the end of Sec. 30). By Theorem 32.2, M is a Warfield module. 3317
Thus, countably generated modules with decomposition bases are determined by their Ulm–Kaplansky invariants and Warfield invariants. Any module of rank 1 necessarily has a decomposition basis. Consequently, the property of definability by invariants holds for countably generated modules of rank 1. This has been proved by Kaplansky and Mackey [163], Rotman [255], and Megibben [224]. The paper of Rotman and Yen [257] contains the existence theorems for such modules. In Sec. 36, we will need the following result of Files [71]. Lemma 32.10. If M is a Warfield module and λ = l(t(M )), then M has a nice decomposition basis X such that M/RX is a totally projective module and pλ M ⊆ RX. Proof. There exists a totally projective module T such that M ⊕ T is a simply presented module (Corollary 32.3). In addition, we can choose T with pλ T = 0. Indeed, it follows from the proof of Theorem 32.2 (2) that the role of T can beplayed by the module T /pλ T , which is also simply presented (Sec. 30, property (f)). Thus, M ⊕ T = Mi , where all Mi are simply presented modules of rank 1. We i∈I
have
pλ (M ⊕ T ) = pλ M =
pλ Mi ,
i∈I
∼ where, for every i ∈ I, either i = 0 or i = R. If the first case holds, then we take an element xi of infinite order in Mi such that {xi } is a nice basis (see Theorem 32.2 (1)). In addition, we can assume that xi ∈ M . In the second case, let pλ Mi = Rxi . The factor module Mi /Rxi also is totally projective by property Mi , (f) from Sec. 30. As a result, we have the nice decomposition basis X = {xi | i ∈ I} for where Mi /RX is a totally projective module. However, X ⊂ M . Consequently, X is a nice basis of the module M and M/RX is a totally projective module, since it is a direct summand of a totally projective module. By construction, pλ M ⊆ RX. pλ M
pλ M
Warfield invariants g(e, M ) have two shortcomings. They are defined for a restricted class of modules. In addition, it is difficult to prove that they, in fact, are “invariants” (the proof uses serious categorytheoretical arguments). Stanton [270] defined new invariants for every module. These invariants coincide with g(e, M ) if g(e, M ) are defined; the invariants have no such shortcomings. Let u = {σi }i≥0 be a height sequence such that σi = ∞ for all i. Recall that if M is a module, then M (u) = {x ∈ M | U (x) ≥ u} (see Sec. 28). Now we denote by M ∗ (u) the submodule generated by all elements x ∈ M (u) such that h(pi x) > σi for infinitely many subscripts i. The modules M (u) and M ∗ (u) are completely invariant submodules of the module M . It follows from the definition that pM (u) ⊆ M ∗ (u), and, therefore, M (u)/M ∗ (u) is a space over the residue field Rp . In Sec. 28, we have defined the height sequence pi u = (σi , σi+1 , . . .) for every i ≥ 0. The sequences u and pi u are contained in the same equivalence class of height sequences. Lemma 32.11. The mapping ϕi : M (u)/M ∗ (u) → M (pi u)/M ∗ (pi u) defined by the relation
ϕi (x + M ∗ (u)) = pi x + M ∗ (pi u)
is a monomorphism for every i ≥ 0. Proof. Since x ∈ M ∗ (u) implies pi x ∈ M ∗ (pi u), ϕi is well defined; it is clear that ϕi is a homomorphism. We show that ϕ1 is a monomorphism. In this case, every ϕi is also a monomorphism, since it is a n rk xk , where all xk are composition of monomorphisms. Let x + M ∗ (u) ∈ Ker(ϕ1 ). Then px = contained in the generator system for M ∗ (pu) indicated in the definition. Let U (xk ) = (τk0 , τk1 , . . .). 3318
k=1
Then τkj ≥ σj+1 for every j. For every k, there exist infinitely many subscripts j with τkj > σj+1 . We have xk = pyk , where theelement yk has the height sequence (ρk , τk0 , τk1 , . . .) with ρk ≥ σ0 . Therefore, yk ∈ M ∗ (u) and p(x rk yk ) = 0. The elements of finite order in M (u) are contained in M ∗ (u). − ∗ ∗ Consequently, x − rk yk ∈ M (u) and x ∈ M (u), since rk yk ∈ M ∗ (u). Assuming that the monomorphisms ϕi are embeddings for convenience, we obtain the following ascending chain of Rp -spaces: M (u)/M ∗ (u) ⊆ M (pu)/M ∗ (pu) ⊆ · · · . For a more accurate presentation, we can consider the following direct spectrum of Rp -spaces: {M (pi u)/M ∗ (pi u); ϕij (i, j ≥ 0)}, where the monomorphisms ϕij are meaning. (Direct spectra and their limits are defined of ani obvious ∗ i in Sec. 29.) We take the union M (p u)/M (p u) or, equivalently, the limit of our direct spectrum. i≥0
We call attention to the following detail. Assume that the height sequence v is equivalent to u. Then pm v = pn u for some integers m, n ≥ 0. Therefore, it is clear that we can substitute v for u in the above union and then nothing changes. We conclude that the union depends only on the equivalence class e containing the sequences u and v, but the union does not depend on the sequences u, v, . . . themselves. We denote by Se (M ) the union considered. Let h(e, M ) be the dimension of Se (M ) over the field Rp . Thus, we have the invariant h(e, M ) of the module M . The cardinal numbers h(e, M ) for all possible classes e are called Stanton invariants of the module M . Lemma 32.12. (a) Let M be a module of rank 1, and let e be an equivalence class of height sequences. Then h(e, M ) = 1 for e = U (M ) and h(e, M ) = 0 for e = U (M ). (b) If M = Mi , then h(e, M ) = h(e, Mi ) for every class e. i∈I
i∈I
Proof. (a) Choose an element x ∈ M of infinite order. Let U (x) = u = (σ0 , σ1 , . . .) ∈ e; therefore, e = U (M ). Let y ∈ M (u) \ M ∗ (u) and let U (y) = (τ0 , τ1 , . . .). There exists a subscript j ≥ 0 such that τi = σi for all i ≥ j. Therefore, there exist an invertible element s ∈ R and an integer n ≥ 0 such that pn sx = pn y (since M has rank 1). Consequently, the order of the element y − sx is finite. Therefore, y − sx ∈ M ∗ (u) and y + M ∗ (u) = sx + M ∗ (u). This proves that M (u)/M ∗ (u) is a one-dimensional Rp -space. In other words, h(e, M ) = 1. Now we assume that e = U (M ) and v = (ρ0 , ρ1 , . . .) ∈ e. Let z ∈ M (v) \ M ∗ (v). Then almost all coordinates of the sequence U (z) coincide with the corresponding ρi . Consequently, U (z) and v are equivalent to one another, which is impossible. Thus, in fact, we have M (v) = M ∗ (v) and h(e, M ) = 0. (b) We take some sequence u ∈ e. The relations Mi (u) and M ∗ (u) = Mi∗ (u) M (u) = i∈I
i∈I
imply the canonical isomorphism M (u)/M ∗ (u) ∼ =
Mi (u)/Mi∗ (u).
i∈I
It follows from such isomorphisms that Se (M ) ∼ =
Se (Mi ),
i∈I
which implies the required equality for invariants. Corollary 32.13 (Stanton [270]). Let M be a direct summand of a completely decomposable module. Then g(e, M ) = h(e, M ) for every equivalence class e of height sequences. 3319
Proof. First, we consider the following circumstance. If T is a primary module, then T (u) = T ∗ (u) for every sequence u ∈ e. Therefore, h(e, T ) = 0 and h(e, M ⊕T ) = h(e, M ), where M is an arbitrary module. Consequently, Stanton invariants are calculated up to primary direct summands. By assumption, we have M ⊕ N = Ai , where N is some module and all Ai are modules of rank 1. i∈I
It follows from Corollary 29.4 that for some primary modules S and T , we obtain Ai ⊕ S, M ⊕T ∼ = i∈J
where J ⊆ I. Furthermore, we consider Lemma 32.12 (b) and obtain h(e, M ) =
i∈J
h(e, Ai ). On the other
hand, it follows from the definition of Warfield invariants that Ai = g(e, Ai ). g(e, M ) = g e, i∈J
i∈J
Lemma 32.12 (a) completes the proof. Finally, we show that Warfield modules have some natural projectivity property. We recommend comparing the following lemma with Lemma 10.1 (1) in Part I. We recall that h-balanced exact sequences were introduced in Sec. 28. Lemma 32.14. For every module M , there exists an h-balanced exact sequence 0 → L → P → M → 0, where P is a simply presented module. Proof. Let X be a set such that the cardinality of X is equal to the cardinality of M , and let θ : X → M be some bijection. Let P be a module with a generator system X and defining relations of the form px = 0 and px = y (x, y ∈ X). In addition, we assume that px = 0 (resp., px = y) iff the relation pθ(x) = 0 (resp., pθ(x) = θ(y)) holds in M . (See the beginning of Sec. 30 on the existence of such a module P .) The module M satisfies the definition of a simply presented module. The mapping θ extends to an epimorphism ϕ : P → M in the standard way. We set L = Ker(ϕ). It remains to prove that the obtained sequence is an h-balanced exact sequence. We use Proposition 28.7. By induction, we can verify that the height in P of every element x of X is equal to the height of the element θ(x) in M . By this property, we can assume that U (x) = U (θ(x)) for every x ∈ X. Therefore, we obtain that ϕ(P (u)) = M (u) for every height sequence u. We also need to prove that ϕ((pσ P )[p]) = (pσ M )[p] for all ordinal numbers σ. If x ∈ X and the order of the element ϕ(x) is equal to p, then the same is true for x. Consequently, U (x) = U (ϕ(x)). Now it is clear that the required relation is a consequence of the definition of the module P . Theorem 32.15 (Warfield [299, 305]). A module M is projective with respect to any h-balanced exact sequence iff M is a Warfield module. Proof. Let the module M be projective with respect to every h-balanced exact sequence. This means that for every epimorphism π : B → C with h-balanced submodule Ker(π) and a homomorphism ϕ : M → C, there exists a homomorphism ψ : M → B with ϕ = ψπ (for comparison, see the end of Sec. 10 (Part I) on the definition of purely projective modules). By Lemma 32.14, there exists an epimorphism π : P → M , where P is a simply presented module and Ker(π) is an h-balanced submodule in P . Since the module M is projective with respect to any h-balanced exact sequence, there exists a homomorphism χ : M → P with χπ = 1M . Therefore, P = Ker(π) ⊕ M , where M ∼ = M (see the proof of (1) =⇒ (3) of Theorem 5.4 Part I). Thus, M is a Warfield module. 3320
Now we assume that M is a Warfield module and π
0→A→B→C→0
(1)
is an h-balanced exact sequence. The module M is a direct summand of some simply presented module. Clearly, we can assume that M is simply presented. Such a module is a direct sum of a totally projective module and simply presented modules of rank 1 (Theorem 30.3). Using a straightforward argument, it can be verified that any direct sum modules that are projective in our sense is projective. Consequently, it suffices to prove the assertion when M is a totally projective module or a simply presented module of rank 1. Any totally projective module is projective with respect to every balanced exact sequence of primary modules (Theorem 30.4). Assume that we have an h-balanced exact sequence (1). Using Proposition 28.7, we can verify that the induced sequence of torsion submodules 0 → t(A) → t(B) → t(C) → 0 is an h-balanced exact sequence; therefore, it is a balanced exact sequence (see the end of Sec. 28 and Proposition 28.8). Now it is clear that a totally projective module is projective with respect to sequence (1). Finally, let M be a mixed, simply presented module of rank 1. There exists an element x ∈ M of infinite order such that M/Rx is a totally projective module (Corollary 32.8 (b)). We assume that ϕ : M → C is some homomorphism. Since π(B(U (x))) = C(U (x)) by Proposition 28.7, there exists y ∈ B(U (x)) with the properties π(y) = ϕ(x) and U (x) ≤ U (y). The mapping ψ : Rx → Ry that maps x into y is a homomorphism not decreasing the heights. By Corollary 32.8 (a), ψ extends to a homomorphism M → B, which is also denoted by ψ. We take the homomorphism ϕ − ψπ : M → C. Since the submodule Rx is contained in its kernel, we can consider ϕ − ψπ as a homomorphism M/Rx → C. We noted earlier that a totally projective module is projective with respect to the sequence (1). This means that there exists ξ : M → B such that Rx ⊆ Ker(ξ) and ξπ = ϕ − ψπ. Therefore ϕ = (ψ + ξ)π, which is required. In the proof of the theorem, it is mentioned that the class of balanced exact sequences is larger than the class of h-balanced exact sequences. The modules projective with respect to every balanced exact sequence compose a very interesting subclass of the class of Warfield modules, which is rich in content. The theory of such modules was developed by Warfield in [302]. Exercise 1 (Hunter–Richman–Walker [152]). If M is a Warfield module, then any direct sum of infinitely many copies of the module M is a simply presented module. Exercise 2 (Jarisch–Mutzbauer–Toubassi [158]). Let M be a module such that the torsion submodule of M is a direct sum of cyclic modules. Then M is a Warfield module iff pn M is a Warfield module for some n ≥ 1. Exercise 3. Transfer Corollary 32.9 to direct sums of countably generated modules. Exercise 4. A countably generated module M is a Warfield module iff there exists a countably generated primary module T such that M ⊕ T is a simply presented module. Exercise 5 (Files [70]). (a) An exact module of rank > 1 is not a Warfield module. (b) If two exact modules have the same Ulm–Kaplansky invariants and the modules are isomorphic to one another in Warf, then they are isomorphic. Remarks. There exist different approaches to the study of mixed modules and Abelian groups. They can be considered as extensions of torsion modules by torsion-free modules (for example, by divisible modules). The diametrically opposed approach is to study these objects as extensions of torsion-free modules (for example, free) by using torsion modules. Other ways combine these two approaches and represent mixed modules as couniversal squares (see Schultz [266, 267]) or construct them by the use of torsion-free modules (see Sec. 22 (Part I)). 3321
A review of studies related to Ulm’s theorem is presented in Sec. 30. Furthermore, the paragraph before Lemma 32.10 contains some historical remarks on generalizations of the Ulm theorem to countably generated modules of rank 1. Mackey, a student of Kaplansky, obtained a short proof of the Ulm theorem; his argument was published in [160] and was essential for subsequent generalizations of the Ulm theorem. Rotman and Yen (see [255, 257]) and Bang (see [28–30]) extended the mentioned results on countably generated modules of rank 1 to countably generated modules of finite rank and their direct sums. These papers also contain other results on such modules (see also Stratton [275]); in particular, the paper [257] contains the existence theorem (similar theorems are discussed below). As for the theory presented here, it is necessary to note that activity in this field proceeded in parallel for modules and for p-local Abelian groups (i.e., Zp -modules, see Sec. 4 (Part I)). Formally, the activity frequently proceeded for Zp -modules. However, the results on Zp -modules usually did not require changes (only the corresponding specifications) for the extension to modules over an arbitrary discrete valuation domain. This property was already mentioned in Kaplansky’s work [160]. In [299], Warfield pointed cut possibility of generalization of the main results on totally projective Abelian groups to primary modules. In general, we follow the ideas of Warfield. His papers [299, 300, 302, 303] and [305] have given a powerful impetus to the studies of mixed modules and Abelian groups. An alternative method of the proof of old and new (at that time) results was developed by Hunter, Richman and Walker in [151] and [152]. They used ideas from the theory of groups with valuations. Hill and Megibben [143] presented a completely different method, which can be called combinatorial. They do not use any category-theoretical methods in their proofs. Their methods also have other advantages. Jarisch, Mutzbauer, and Toubassi ([156–158]) obtained several results on simply presented modules and Warfield modules M such that t(M ) is the direct sum of cyclic modules and M/t(M ) is a divisible module. If we need to deal with Warfield Zp -modules, then we also speak about the local case and local Warfield groups. The theory of local Warfield groups was extended to Abelian groups, which are not necessarily Zp -modules. Assuming this common theory, we speak about the global case and the global Warfield groups. The theory of global Warfield groups finds its origin in the Ulm–Zippin theory of countable p-groups and the Baer theory of completely decomposable torsion-free groups. These two theories were united by the notion of a simply presented group, which was introduced for p-groups by Crawley and Hales in [50]; the notion was transferred to arbitrary groups in [300]. Global Warfield groups were originally defined as direct summands of simply presented groups. The study of such groups was initiated by the work of Warfield [300]. Later on, the papers of Stanton [271, 274], Hunter and Richman [150], Hill and Megibben [145], and Hunter, Richman and Walker [152] appeared. For some aspects of this theory, the theorem of Azumaya type for an additive category proved by Arnold, Hunter, and Richman in [15] is very useful. Several topics related to local and global Warfield groups are presented in Loth’s wark [206]. We did not touch on the independence of systems of invariants f (σ, M ) and g(e, M ) for Warfield modules (this notion of independence was introduced in Sec. 6 (Part I)). For countable Abelian p-groups, there is the Zippin existence theorem (this theorem is mentioned at the end of Sec. 30). In general, an existence theorem asserts that there exists a certain module with given invariants (in our situation, the theorem asserts that there exists a Warfield module with given Ulm–Kaplansky invariants and Warfield invariants). In the papers of Hunter, Richman and Walker [150, 152], the existence theorems for local and global Warfield groups are proved. The torsion submodule of a Warfield module is not necessarily a totally projective module. Torsion subgroups of local Warfield groups are called S-groups. Their description is contained in papers of Warfield [301], Hunter [149], Stanton (preprint), and Hunter and Walker [153]. In particular, [301] contains invariants for S-groups. Returning to the beginning of these remarks, it is possible to say that we prefer the second approach to the studies of mixed modules. In connection with the first-mentioned approach to mixed modules, we emphasize a very interesting paper of Mutzbauer and Toubassi [235] on mixed modules M such that t(M ) is the direct sum of cyclic modules and M/t(M ) is a divisible module (see also [156–158]). 3322
Problem 13. (a) Which modules M satisfy the property that every pure closed submodule of M is a direct summand of M ? (Paper [42] is related to this topic.) (b) Which modules M satisfy the property that every balanced submodule of M is a direct summand of M ? Problem 14. (a) Describe modules M and N such that HomW (M, N ) = HomW (M, N ) (see Corollary 32.4). (b) Find conditions implying that modules isomorphic to each other in Warf are isomorphic in Walk. (c) Find conditions that imply that the indecomposability of the module M in Walk is equivalent to the indecomposability of M in Warf. (Corollary 32.4 is related to (b) and (c).) The following two problems are contained in the list of 27 problems formulated by Warfield in [303]; some of the problems are still open. Problem 15. Which R-algebras are endomorphism algebras in Walk or Warf of mixed modules? In particular, answer the question in the case where R is a complete domain or the modules are of finite rank (see Corollary 22.3 in Part I). More precisely, obtain characterization theorems or realization theorems in the sense of Chapter 4 (Part I) for endomorphism rings (algebras) in Walk or Warf of mixed modules. In the papers of Bang [29, 30], in fact, it was proved that for direct sums of countably generated modules over a complete discrete valuation domain, M ∼ = N iff M and N have equal Ulm–Kaplansky invariants and they are isomorphic to one another in Warf. (Stratton [275] proved that the completeness is essential in this result.) Similar results holds for Warfield modules and exact modules (Corollary 32.7 and Exercise 6 in Sec. 32). Problem 16. Find classes of modules satisfying the following property. For any two modules M and N in such a class, if M and N have equal Ulm–Kaplansky invariants and they are isomorphic to one another in Warf (or Walk), then M and N are isomorphic. Problem 17. In the sense of [177, Secs. 32, 34], study relations between some subcategories of the category Walk or Warf and some subcategories of left modules over the endomorphism ring of the mixed module A in Walk or Warf. For example, prove the equivalence between the categories. Taking A as a module of rank 1, try to prove Corollary 29.4 and Corollary 29.9 using these equivalences and without using Sec. 27. Probably, it is possible to use [302, Theorem 3.11]. ([303, Problem 14] is related to this problem.) Problem 18. Construct the theory of mixed modules that are extensions of totally projective modules by divisible torsion-free modules (of finite rank). (It should be noted that there is an important paper [235].)
Chapter 7 DEFINABILITY OF MODULES BY THEIR ENDOMORPHISM RINGS In Chapter 7, we consider the following topics: theorems of Kaplansky and Wolfson (Sec. 33); theorems of a topological isomorphism (Sec. 34); modules over completions (Sec. 35); endomorphisms of Warfield modules (Sec. 36). In Chapter 4, we were interested in the following question: what can be said about endomorphism rings (or endomorphism algebras) of modules? In this chapter, we consider another fundamental problem on endomorphism rings. The solution of this problem can answer the question on the uniqueness of the 3323
realization of the ring by the endomorphism ring of some module that naturally appeared in Chapter 4. This is the following problem: how much would the endomorphism ring (or the endomorphism algebra) determine the original module? In the classical formulation, the following results solve this problem. If End(M ) ∼ = End(N ), then M ∼ = N , or, more generally, there exists a semilinear isomorphism of the modules M and N . Such theorems are called isomorphism theorems (in the weak sense). The isomorphism theorem in the strong sense usually means that the given isomorphism of endomorphism rings ψ : End(M ) → End(N ) is induced by an isomorphism (or a semilinear isomorphism) ϕ : M → N . The last property means that ψ(α) = ϕ−1 αϕ for every α ∈ End(M ) (see (c) in Sec. 2 (Part I)). This type of isomorphism theorems is related to the following problem: define modules such that all automorphisms of their endomorphism rings are inner. Assuming that endomorphism rings are equipped with the finite topology, we can consider continuous isomorphisms of endomorphism rings. They contain more information on the original modules. In this case, we speak about the theorems on a topological isomorphism. In Sec. 33, we will see that the situation with isomorphism theorems for primary modules and torsionfree modules over a complete domain is very good. In the class of torsion-free modules over an incomplete domain, isomorphism theorems are quite rare. Here, the “nonisomorphism theorems” are more common (see Theorem 19.10 in Part I). In contrast to the case of primary modules, the situation for mixed modules is more complicated (even in the case of mixed modules over a complete domain). It is not absolutely clear which classes of mixed modules can provide isomorphism theorems and which methods are necessary for the proof of such theorems. However, this is not surprising if we consider Theorem 19.10 and Corollary 22.3 (Part I). In Sec. 34, we examine the role of the finite topology in the isomorphism theorems for mixed modules. In Secs. 34 and 35, we obtain sufficiently wide classes of mixed modules in which the isomorphism theorems hold. We also present the feasibility bounds for such theorems. In Sec. 36, we give an affirmative answer to the isomorphism problem in the weak sense for endomorphism algebras of Warfield modules. Since our endomorphism rings are R-algebras, it is natural to consider their algebra isomorphisms. This does not lead to a loss of generality. is the p-adic completion of R. In this chapter, R is a commutative discrete valuation domain and R 33.
Theorems of Kaplansky and Wolfson
We obtain the isomorphism theorems for endomorphism rings of primary modules and torsion-free modules over a complete, discrete-valuation domain. Let A be an algebra over some commutative ring S (algebras are defined in Sec. 19 (Part I)). The mapping s → s · 1A (s ∈ S) is a homomorphism from the ring S into the center of the algebra A. If the homomorphism is an embedding, i.e., A is a faithful S-module, then we identify the elements s and s · 1A and assume that S is a subring of the center of the algebra A. A mapping A → B that is a ring isomorphism and an isomorphism of S-modules is called an isomorphism from the algebra A onto the S-algebra B. Consequently, a ring isomorphism ψ : A → B is an isomorphism of algebras provided ψ(sa) = sψ(a) = (s · 1B )ψ(a) for all s ∈ S and a ∈ A. We also have ψ(sa) = ψ(s · 1A · a) = ψ(s · 1A )ψ(a), whence ψ(s · 1A ) = s · 1B . If A and B are faithful algebras, then ψ(s) = s. Therefore, isomorphisms of faithful S-algebras coincide on elements of S with the identity mapping. Now we take some R-module M . We have the endomorphism R-algebra End(M ) (see the beginning of Sec. 19 (Part I)). The canonical mapping R → End(M ) is not an embedding only if M is a bounded module. Let M be a bounded module and let pk be the least upper bound of the orders of elements of M . Then M is an Rpk -module and End(M ) is an Rpk -algebra (the conversion of an R-module into an Rpk -module is discussed in Sec. 4 (Part I)). According to the above, we identify Rpk with the image of the embedding Rpk → End(M ). If M is an unbounded module (i.e., is not a bounded module), then we 3324
identify R with the image of the embedding R → End(M ). Thus, if M and N are R-modules, then we can consider ring isomorphisms, as well as “stronger” isomorphisms of R-algebras between End(M ) and End(N ). In the study of isomorphism theorems, it is not necessary to assume that the modules considered are modules over the same ring. In such a case, semilinear isomorphisms of modules are used. Let S and T be two rings, M be an S-module, and N be a T -module. An additive isomorphism ϕ : M → N is called a semilinear isomorphism of the modules S M and T N if there exists a ring isomorphism τ : S → T such that ϕ(sa) = τ (s)ϕ(a) for all s ∈ S and a ∈ A. We say that an isomorphism of endomorphism rings (or endomorphism algebras) ψ : EndS (M ) → EndT (N ) is induced by the semilinear isomorphism ϕ : S M → T N provided ψ(α) = ϕ−1 αϕ for every α ∈ End(M ). It should be noted that even for S = T , we only frequently can prove that ψ is induced by some semilinear isomorphism ϕ : S M → S N (but not by an ordinary isomorphism). In this case, τ is some automorphism of the ring S. We always assume that M and N are modules over the same ring R. Except for Theorem 33.2 and several exercises, we consider R-algebra isomorphisms between End(M ) and End(N ). These restrictions do not lead to a considerable loss of generality but free us from technical difficulties. The isomorphisms of endomorphism algebras considered are their R-algebra isomorphisms. We describe the center of the endomorphism ring of a primary module. This is of independent interest and will be used in studying the isomorphisms of endomorphism rings. We note that direct summands of the module are invariant with respect to central endomorphisms. Indeed, let M be a module, A be a direct summand of M , γ ∈ Z(End(M )), and ε : M → A be the projection. Then γA = γ(εM ) = ε(γM ) ⊆ A. Furthermore, if Ra and Rb are cyclic modules of order pm and pn , respectively, then the existence of a homomorphism α : Ra → Rb with αa = b is equivalent to the inequality n ≤ m. It is convenient to consider primary modules as modules over a complete domain. Theorem 33.1 (Kaplansky [160]). Let M be a primary module over a complete discrete valuation domain R. Then Z(End(M )) = Rpk provided the module M is bounded and pk is the least upper bound of the orders of elements of M ; otherwise, Z(End(M )) = R. Proof. It suffices to prove that an arbitrary element γ ∈ Z(End(M )) is contained in the corresponding ring. (a) Assume that the module M is bounded. By Theorem 4.8 (Part I), M is a direct sum of cyclic modules whose orders do not exceed pk , where pk is the element from the theorem. The module M has a cyclic direct summand Ra of order pk . Considering Example 12.2 (Part I), we obtain γa = s¯a, where s¯ ∈ Rpk . For an arbitrary element x ∈ M , there exists an α ∈ End(M ) such that αa = x. We have γx = γ(αa) = α(γa) = α(¯ sa) = s¯x, whence γ = s¯. (b) For a nonbounded module M , we separately consider the case where the module has the form M = A ⊕ D, where A is a bounded module and D is a nonzero divisible module. Let Ra be a cyclic direct summand of the module A that is maximal among the elements of the module A of order pk . It follows from (a) that the restriction of the endomorphism γ to A coincides with the multiplication by some element s¯ ∈ Rpk . We denote by E some direct summand of the module D isomorphic to R(p∞ ). By Example 12.5 (Part I), the restriction of the endomorphism γ to E coincides with the multiplication by some element r ∈ R. For an arbitrary element y ∈ D, there exist e ∈ E and β ∈ End(M ) with y = βe. Therefore, γy = γ(βe) = β(γe) = β(re) = r(βe) = ry. 3325
This relation means that γ coincides with r on the whole module D. Now we choose an element e ∈ E of order pk and the endomorphism ξ ∈ End(M ) with ξa = e. We have re = γ(ξa) = ξ(γa) = ξ(¯ sa) = s¯e. Consequently, (r − s¯)e = 0 and s¯ = r¯, where r¯ = r + pk R. We can state that γ is the multiplication by r on the whole module M . Thus, γ = r. It remains to consider the case where M = A ⊕ D and the module A is not a bounded module. Basis submodules of the module M also are not bounded (see Theorem 7.2 (Part I)). By Corollary 9.3 (Part I), there exist decompositions M = Ra1 ⊕ · · · ⊕ Rai ⊕ Mi , i = 1, 2, . . . , such that Mi = Rai+1 ⊕ Mi+1 and the orders pni of elements ai satisfy 1 ≤ n1 ≤ · · · ≤ ni ≤ · · · . For every pair of subscripts i and j with i < j, we take some endomorphism εji of the module M that maps aj into ai . Then γai = s¯i ai = si ai for all i, where s¯i = si + pni R. If i < j, then it follows from si ai = γai = (εji γ)aj = (γεji )aj = εji sj aj = sj ai that sj − si ∈ pni R. Consequently, {si }i≥1 is a Cauchy sequence in R. Taking the limit s of the sequence, we obtain sai = si ai for every i. Let x be some element of the module M . There are an element ai and an endomorphism α ∈ End(M ) such that αai = x. Furthermore, γx = γαai = αγai = αsai = sx and γ = s.
Our first isomorphism theorem is related to primary modules. Under a ring isomorphism, the center is mapped into the center. Therefore, it follows from Theorem 33.1 that we can assume that the modules M and N from Theorem 33.2 are modules over the same ring. Similarly to Theorem 33.1, it is convenient to distinguish the case of the bounded module M and the case of the unbounded module M . Let M be a bounded module and let pk be the least upper bound of the orders of elements of M . If N is some R-module and End(M ) ∼ = End(N ), then it follows from pk M = 0 that pk End(M ) = 0 = pk End(N ) and k p N = 0. Consequently, N is a bounded module. It is clear that pk is the least upper bound of the orders of elements of the module N . It is sufficient to consider only two cases in the theorem. Theorem 33.2 (Kaplansky [160]). Let R be a complete discrete-valuation domain, and let M and N be either Rpk -modules or nonbounded primary R-modules. Then every ring isomorphism between End(M ) and End(N ) is induced by some semilinear isomorphism between M and N . Proof. First, we note that the following property holds. Let M be a cyclic primary module, N be some primary module, and let End(M ) ∼ = End(N ). Then N is an indecomposable module (property (a) from Sec. 2 (Part I)); therefore, N is a cyclic or quasicyclic module (Corollary 7.2 (Part I)). Using Examples 12.2 and 12.5 (Part I), we obtain M ∼ = N. We begin with some ring isomorphism ψ : End(M ) → End(N ). For α ∈ End(M ), we write ψ(α) = α∗ . We use some notation and results from the proof of Theorem 33.1. Let M and N be two Rpk -modules. It follows from Theorem 33.1 that ψ induces the automorphism r¯ → r¯∗ , r¯ ∈ Rpk , of the ring Rpk . The modules M and N are direct sums of cyclic modules. Let a be a generator of a cyclic direct summand module M of maximal order pk . If ε : M → Ra is the projection, then ε is an idempotent of the ring End(M ) and ε∗ is an idempotent of the ring End(N ). Consequently, ε∗ N is a direct summand of the module N . By property (d) from Sec. 2 (Part I), ψ induces the ring isomorphism End(Ra) → End(ε∗ N ). By the above remark, ε∗ N is a cyclic module Rb of order pk . For an arbitrary element x ∈ M , we choose an endomorphism α of the module M with x = αa. We define ϕ : M → N by the relation ϕx = α∗ b. 3326
This definition is correct, i.e., it does not depend on the choice of the endomorphism α. If x = α1 a, where α1 ∈ End(M ), then (α − α1 )a = 0 and ε(α − α1 ) = 0. Therefore, (ε(α − α1 ))∗ = ε∗ (α∗ − α1∗ ) = 0 and
(α∗ − α1∗ )b = 0.
We take one more element y ∈ M and choose β ∈ End(M ) with y = βa. Then x + y = (α + β)a and ϕ(x + y) = (α + β)∗ b = α∗ b + β ∗ b = ϕx + ϕy, i.e., ϕ preserves the operation of addition. For every element r ∈ R, we have rx = r(αa) = (rα)a. Therefore, ϕ(rx) = (rα)∗ b = r∗ α∗ b = r∗ ϕ(x). If ϕx = α∗ b = 0, then (εα)∗ = ε∗ α∗ = 0. Therefore, εα = 0, x = εαa = 0, and Ker(ϕ) = 0. The module Rb is a direct summand in N of maximal order. Consequently, for every element z ∈ N , there exists an endomorphism δ ∈ End(N ) with z = δb. We have δ = α∗ for some α ∈ End(M ). Then z = δb = α∗ b = ϕx, where x = αa. We obtain that ϕ is a bijection; in other words, ϕ is a semilinear isomorphism. For μ ∈ End(M ), we represent the element z = ϕx in the form z = α∗ b for some α ∈ End(M ). Now we have μ∗ z = (αμ)∗ b = ϕ((αμ)a) = ϕ(μx) = ϕ(μ(ϕ−1 z)) = (ϕ−1 μϕ)z, i.e., ψ(μ) = μ∗ = ϕ−1 μϕ. Therefore, ϕ induces ψ. Now let M and N be unbounded primary R-modules. The isomorphism ψ induces the automorphism r → r∗ , r ∈ R, of the ring R (Theorem 33.1). If M is a quasicyclic module, then (similarly to the beginning of the proof) we can prove that N is a quasicyclic module and M ∼ = N. We partition the remaining part of the proof into two parts. 1. The module M has the form M = A ⊕ D, where A is a bounded module and D is a nonzero divisible module. Let Ra be a cyclic direct summand of maximal order pk of the module A, E be a quasicyclic direct summand of the module D, and let c1 , . . . , cn , . . . be a generator system of the module E such that pc1 = 0 and pcn+1 = cn for n ≥ 1. We denote by ε : M → Ra and π : M → E the corresponding projections. Similarly to the first part of the proof, we obtain that ε∗ N is a cyclic direct summand of order pk of the module N , and, by the above, π ∗ N is a quasicyclic direct summand of the module N . Let ε∗ N = Rb, and let d1 , . . . , dn , . . . be a generator system of the module π ∗ N such that pd1 = 0 and pdn+1 = dn for n ≥ 1. We represent an arbitrary element x ∈ M in the form x = x1 + x2 , where x1 ∈ A and x2 ∈ D, and we take an endomorphism α of the module M such that αa = x1 and α1 cm = x2 . In addition, we can assume that m ≥ n. Then ε(α − α1 ) = 0 and (pm−n α − α1 )cm = 0. Consequently, the endomorphism π(pm−n α − α1 ) annihilates E[pm ]. It follows from Example 12.5 (Part I) that the endomorphism is divisible by pm . Then the endomorphism (π(pm−n α − α1 ))∗ is also divisible by pm . Consequently, it annihilates the element dm . We obtain α∗ b = α1∗ b, α∗ dn = pm−n α∗ dm = α1∗ dm , and α∗ (b + dn ) = α1∗ (b + dm ). Similarly to the case of bounded modules, we can verify that ϕ is a semilinear isomorphism inducing ψ. 2. Let M = A ⊕ D, where the module A is an unbounded module. We take direct decompositions of the module M presented in the proof of Theorem 33.1. Let εi be the projection M → Rai . For subscripts i < j, we define the endomorphism εji of the module M that maps aj into ai and annihilates the complement summand for Raj ; also, we define the endomorphism εij that maps ai into pnj −ni aj and annihilates the complement to Rai . In this case, we have the following: (1) εi are pairwise orthogonal idempotents; (2) εi εij = εij εj = εij ; (3) εij εji = p|nj −ni | εi ; (4) εij εjk = εik if i < j < k or i > j > k. The submodules ε∗i N are cyclic direct summands of the module N . It follows from (2) that ε∗i+1,i maps from ε∗i+1 N into ε∗i N . We set ε∗ N = Rbi and show that it is possible to choose generators bi such that ε∗i+1,i bi+1 = bi for all i. If b1 , . . . , bi are already chosen and the element ei+1 generates the submodule 3327
ε∗i+1 N , then ε∗i+1,i ei = sbi for some s ∈ R. By (3), we obtain ε∗i,i+1 sbi = pni+1 −ni ei+1 . Considering the orders of elements, we obtain that s is an invertible element of the ring R. We take the element bi+1 = s−1 ei+1 . Then ε∗i+1,i bi+1 = bi . By property (4), ε∗ji bj = bi for all i < j. For x ∈ M , we choose an endomorphism α ∈ End(M ) such that αai = x for some i. Let ϕ : x → α∗ bi . If α1 aj = x and j ≥ i, then εj (εji α − α1 ) = 0. Therefore, ε∗j (ε∗ji α∗ − α1∗ ) = 0, which means that α∗ bi = α1∗ bj . Therefore, the mapping ϕ is well defined. Once again, we can verify that ϕ is a semilinear isomorphism inducing ψ. Let ψ : End(M ) → End(N ) be a ring isomorphism for the primary R-modules M and N . In the beginning of the section, it is noted that ψ is an isomorphism of R-algebras provided ψ(r) = r for all r ∈ R. It follows from Theorem 33.1 that isomorphisms between the algebras End(M ) and End(N ) coincide with ring isomorphisms that do not move elements of the center. To formulate an interesting corollary of Theorem 33.2, we set M = N . Let ψ be some automorphism of the ring End(M ). We assume that ψ is induced by some automorphism ϕ of the module M , i.e., ψ(α) = ϕ−1 αϕ for every α ∈ End(M ). Since automorphisms of the module M are invertible elements of the ring End(M ), the last relation means that ψ is an inner automorphism of the ring and that of the algebra End(M ). Corollary 33.3. Let M and N be two primary modules. Then every isomorphism of the algebras End(M ) → End(N ) is induced by some isomorphism M → N . Every automorphism of the endomorphism algebra of a primary module is an inner automorphism. The method used in the proof of Theorem 33.2 is called the Kaplansky method. It has the following meaning. Primitive idempotents of the endomorphism ring of a primary module correspond to direct summands of the module that are isomorphic to R(pk ) or R(p∞ ). To construct an isomorphism from the module M onto the module N , Kaplansky used the transfer of properties of such summands by endomorphisms in order to obtain the required elements of the module N . For torsion-free modules, some complete results are also obtained. (We will use the property of such modules indicated in Corollary 11.7 (Part I).) We consider the main case of reduced modules (see Exercise 6). For a torsion-free module M , the canonical mapping R → End(M ) is an embedding, and we assume that R is a subring in End(M ). Exercises 4, 5, and 7 are related to the following result. Lemma 33.4. Let M be a reduced torsion-free module over a complete, discrete-valuation domain R. Then the center of the ring End(M ) is equal to R. Proof. Let γ ∈ Z(End(M )). We choose some direct summand A of the module M isomorphic to R. Then γA ⊆ A, and γ coincides on A with the multiplication by some element s ∈ R (Example 12.1 in Part I). For an element x ∈ M , there exist a ∈ A and α ∈ End(M ) with αa = x. We have γx = γ(αa) = α(γa) = α(sa) = sx and γ = s. For torsion-free modules, we consider only isomorphisms of endomorphism algebras. Similarly to the case of primary modules, the difficulties appearing in the study of ring isomorphisms are not very interesting (see Exercise 6). Theorem 33.5 (Wolfson [310]). Let M and N be two reduced torsion-free modules over a complete discrete valuation domain R. Then every isomorphism of the algebras End(M ) and End(N ) is induced by some isomorphism between the modules M and N . Every automorphism of the algebra End(M ) is an inner automorphism. Proof. Let ψ : End(M ) → End(N ) be an isomorphism of algebras, and let ψ(α) = α∗ for α ∈ End(M ). We use the Kaplansky method. We fix some direct summand Ra of the module M that is isomorphic to R. Let ε : M → Ra be a projection. Then ε∗ : N → ε∗ N is a projection and End(ε∗ N ) ∼ = End(εN ) ∼ = R. 3328
Consequently, ε∗ N is an indecomposable module and ε∗ N ∼ = R. We have ε∗ N = Rb. Define ϕ : M → N as follows. Let x ∈ M , and let x = αa, where α is an endomorphism of the module M . We set ϕx = α∗ b. Similarly to Theorem 33.2, we obtain that ϕ is an isomorphism of modules M and N that induces ψ. Theorem 19.10 (Part I) leads to the idea that for torsion-free modules over an incomplete domain R, isomorphism theorems are possible only in some sufficiently restricted classes of modules (for example, see Exercise 8 in Sec. 37). By Theorem 16.1 (Part I), End(M ) is a complete topological ring with respect to the finite topology. If the ring R is equipped with the discrete topology, then End(M ) is a complete topological R-algebra. This circumstance is very useful in searching for isomorphism theorems. Precisely, we can consider isomorphisms of endomorphism rings (or endomorphism algebras) continuous in two directions (with respect to finite topologies). Such isomorphisms are called topological isomorphisms. Thus, we say that a ring (resp., algebra) isomorphism ψ : End(M ) → End(N ) is a topological isomorphism of endomorphism rings (resp., endomorphism algebras) if ψ and ψ −1 are continuous with respect to finite topologies on End(M ) and End(N ), respectively. Every isomorphism ϕ : M → N induces the topological isomorphism ψ : End(M ) → End(N ),
ψ(α) = ϕ−1 αϕ,
α ∈ End(M ),
of endomorphism algebras; ψ is an isomorphism of algebras (see property (c) from Sec. 2 (Part I)). Now we take an arbitrary neighborhood of zero UY in the ring End(N ), where Y is some finite subset of the module N , and UY = {β ∈ End(N ) | βY = 0} (see Sec. 16 in Part I). Then ψ −1 UY = Uϕ−1 Y , where Uϕ−1 Y = {α ∈ End(M ) | α(ϕ−1 Y ) = 0} is a neighborhood of zero in the ring End(M ). Conversely, ψ maps neighborhoods of zero of the ring End(M ) onto neighborhoods of zero of the ring End(N ). Consequently, ψ is a topological isomorphism. Topological isomorphisms more completely reflect the structure of a module. The examples of the show that primary modules cannot be distinguished from torsion-free modules by modules R(p∞ ) and R ordinary isomorphisms of endomorphism rings in the general case (see also the paragraph after the proof of Theorem 33.6). In Theorem 33.2 and Theorem 33.5, the finite topology is not explicitly present. However, it follows from Proposition 16.2 (Part I) that every isomorphism End(M ) → End(N ) is continuous for reduced primary modules or torsion-free modules M and N . There are topological variants of Theorem 33.2 and Theorem 33.5 that extend these assertions. In the following theorem, we extend Theorem 33.2 to mixed modules and present a unified formulation of both theorems. First, the following useful result holds. Let ε be some idempotent of the algebra End(M ). The canonical isomorphism End(εM ) ∼ = ε End(M )ε, indicated in Sec. 2 (Part I), property (b), is a topological isomorphism if we assume that End(εM ) is equipped with the finite topology and ε End(M )ε is equipped with the topology induced by the finite topology of the algebra End(M ). Theorem 33.6. (1) Let M and N be two modules over a discrete-valuation domain R and let ψ : End(M ) → End(N ) be a topological isomorphism of algebras. Then there exists an isomorphism ϕ : t(M ) → t(N ) such that ψ(α) and ϕ−1 αϕ coincide on t(N ) for every α ∈ End(M ). (2) Let M be either a primary module or a reduced torsion-free module over a complete discretevaluation domain R and let N be an arbitrary R-module. Then every topological isomorphism of algebras End(M ) ∼ = End(N ) is induced by some isomorphism M ∼ = N. Proof. (1) The required isomorphism ϕ can be constructed by using the Kaplansky method (we can consider only the torsion submodules of the modules M and N ). We only need to recall the following property. If ε is a primitive idempotent of the ring End(M ) and εM ∼ = R(pk ), then it is clear that ψ(ε)N ∼ = k ∞ ∼ ∼ ∼ R(p ). Let εM = R(p ). Then End(εM ) = R (Example 12.5 (Part I)). Consequently, End(ψ(ε)N ) = R. We assume that ψ(ε)N is a torsion-free module. Since ψ(ε)N can be considered as a module over the 3329
(Example 12.1 (Part I)). endomorphism ring of it, ψ(ε)N is an R-module. Consequently, ψ(ε)N ∼ = R By the property mentioned before the theorem, the rings End(εM ) and End(ψ(ε)N ) are topologically isomorphic to one another. However, this is impossible, since the first topology is the p-adic topology and the second topology is discrete. Consequently, ψ(ε)N is a primary module that is isomorphic to R(p∞ ). Now we can use the Kaplansky method. (2) We take some topological isomorphism ψ : End(M ) → End(N ). First, assume that M is a primary module and y is an arbitrary element of the module N . Since ψ is continuous, there exist elements x1 , . . . , xn ∈ M such that ψ(α)y = 0 provided αx1 = · · · = αxn = 0 for some α. There exists a nonzero element r ∈ R such that rxi = 0 for all i. Therefore, 0 = ψ(r)y = ry. Consequently, N is a primary module, and Corollary 33.3 completes the proof. Let M be a reduced torsion-free module. We assume that the module N has a direct summand isomorphic to R(pk ) or R(p∞ ). We consider the isomorphism ψ −1 . It follows from the proof of (1) that the module M also has a similar summand. This is a contradiction. Consequently, N is a reduced torsion-free module. It remains to refer to Theorem 33.5. Earlier, we considered two special cases, where endomorphism algebras are isomorphic to each other but the corresponding modules are not isomorphic. Precisely, if D is a divisible primary module, then the module C = Hom(R(p∞ ), D) is a complete torsion-free module and End(D) ∼ = End(C) (Sec. 13 (Part I)). Let M be an adjusted cotorsion module. Using Sec. 21 (Part I), we can prove that the restriction mapping α → α|t(M ) defines an isomorphism of the algebras End(M ) and End(t(M )). Now it is clear that the both isomorphisms of endomorphism algebras are not topological. Exercise 1 (Baer [27]). Every isomorphism between rings of linear operators of two vector spaces over division rings is induced by a semilinear isomorphism of these spaces. Exercise 2 (Baer [26], Kaplansky [160]). Prove that every isomorphism between endomorphism rings of two primary Abelian groups is induced by some isomorphism between these groups. Exercise 3. Every isomorphism between endomorphism rings of two p-adic, torsion-free modules is induced by some isomorphism of these modules. Exercise 4. Let R be a discrete-valuation domain, and let M be an R-module that has a direct summand isomorphic to R. Then Z(End(M )) = R. Exercise 5. If D is a divisible torsion-free module over a discrete-valuation domain R, then Z(End(D)) ∼ = K, where K is the field of fractions of the domain R. Exercise 6 (Wolfson [310]). Let M and N be two torsion-free modules over a complete discrete-valuation domain R and let ψ : End(M ) → End(N ) be some ring isomorphism. Then both modules M and N are either nondivisible or divisible. In the first case, there exists a semilinear isomorphism of R-modules M → N that induces ψ. In the second case, M and N are vector spaces over the field of fractions K of the domain R and ψ is induced by some semilinear isomorphism of these spaces. Exercise 7. Let R be a discrete-valuation domain, T be a reduced primary R-module, A be a reduced torsion-free R-module, and M = T ⊕ A. (1) If T is a bounded module, then Z(End(M )) = R + pk Z(End(A)), where pk is the least upper bound of the orders of elements of the module T . (2) If T is an unbounded module, then the center Z(End(M )) is canonically isomorphic to the closure ¯ of the subring R in the p-adic topology of the ring Z(End(A)); more precisely, if R ¯∼ R = P , where P is some pure subring in R, then Z(End(M )) consists of multiplications by elements of P . 34.
Theorems of a Topological Isomorphism
We know that End(M ) is a complete (in the finite topology) R-algebra; we simply say “the endomorphism algebra” (see Secs. 19 (Part I) and 33). Up to the end of the chapter, all of the isomorphisms between endomorphism algebras considered are isomorphisms of R-algebras. 3330
We study the problem of a topological isomorphism for endomorphism algebras of mixed modules. We accept the strongest assumptions. Precisely, we assume that the domain R is complete and isomorphisms of endomorphism algebras are topological. We will see that under such assumptions, even for mixed modules of rank 1, two central questions also have a negative answer. We keep in mind the following questions. Does the (topological) isomorphism End(M ) ∼ = End(N ) imply the isomorphism M ∼ = N ? Is every (topological) automorphism of the algebra End(M ) an inner automorphism? We can only prove such that that for every mixed module M , there exists a unique (up to isomorphism over M ) module M ∼ ˜. ∼ M ⊆ M and every topological isomorphism End(M ) = End(N ) is induced by some isomorphism M =N • Clearly, it follows from Theorem 33.6 (1) and properties of the cotorsion hull M that the new “hull” is contained in M • . We present one sufficiently general condition, where M = M , which allows us to M formulate the results on the isomorphism problem in their usual form. Let M be a reduced module. According to Sec. 21 (Part I), we assume that M ⊆ M • . In addition, t(M • ) = t(M ) and M • /M is a divisible torsion-free module. We also use other properties of cotorsion hulls. The group Hom(M, N ) is an R-module, since R is a commutative ring (see Sec. 2 (Part I)). More specifically, if ϕ ∈ Hom(M, N ) and r ∈ R, then the relation (rϕ)(x) = ϕ(rx) (x ∈ M ) defines a homomorphism rϕ ∈ Hom(M, N ) that gives a module multiplication on Hom(M, N ). We extend the notion of the finite topology to homomorphism groups. For a finite subset X ⊆ M , we set UX = {ϕ ∈ Hom(M, N ) | ϕX = 0}. Here, UX is a submodule of the R-module Hom(M, N ). The module Hom(M, N ) is turned into a topological R-module whose basis of neighborhoods of zero consists of the submodules UX for all finite subsets X (we consider the discrete topology on the ring R). The topological R-module Hom(M, N ) is a topological Abelian group such that a module multiplication is continuous with respect to the discrete topology and the finite topology on R and Hom(M, N ), respectively (see also Sec. 11 (Part I) on topological modules). The mapping ψ : Hom(M, N ) → Hom(M , N ) is called a topological isomorphism if ψ and ψ −1 are continuous with respect to finite topology isomorphisms of R-modules (we considered topological isomorphisms of algebras in the previous section). If M ⊆ M , then we have the induced homomorphism of R-modules Hom(M , N ) → Hom(M, N ), ϕ → ϕ|M . It is called the restriction homomorphism. Lemma 34.1 (May [219]). Let M be a reduced module and let t(M ) = T = 0. with properties M ⊆ M and t(M ) = T , and the in(1) There exists a maximal reduced module M , T ) → Hom(M, T ) is a topological isomorphism. Between any two such maximal duced mapping Hom(M modules, there exists a unique isomorphism that coincides with the identity mapping on M . is an R-module, /M is a divisible torsion-free module, and M can be considered as a uniquely (2) M M • defined submodule in M . ); in fact, α of (3) Every α ∈ End(M ) can be uniquely extended to α ˜ ∈ End(M ˜ is the restriction to M • • the unique extension α to α ∈ End(M ). Proof. Let M be some reduced module such that M ⊆ M and t(M ) = T . We define two (possible) properties of the module M . (∗M ) The induced mapping Hom(M , T ) → Hom(M, T ) is a topological isomorphism. (∗ ∗ M ) For every x ∈ M , there exist elements y1 , . . . , yn ∈ M such that β (x) = 0 for every β ∈ Hom(M , T • ) such that β (M ) ⊆ T and β (yi ) = 0 for all i = 1, . . . , n. We assert that (∗M ) holds iff M /M is a divisible torsion-free module and (∗ ∗ M ) holds. Note that the following property holds. For an arbitrary nondivisible module N , there exists a nonzero homomorphism N → R(p). Since T is a primary module, there always exists a nonzero homomorphism N → T . Now we assume that M /M is a nondivisible module. There exists a nonzero homomorphism 3331
¯ Then β¯ : M /M → T . Let β be the composition of the canonical homomorphism M → M /T with β. β ∈ Hom(M , T ) and β = 0, but the restriction of β to M is equal to zero. This contradicts (∗M ). Consequently, M /M is a divisible module. Let px ∈ M , where x ∈ M \ M . Then the height of the element px in M is equal to zero. Otherwise, M \ M has torsion elements, which is impossible, since t(M ) = t(M ). Consequently, there exists a homomorphism β : M → T such that the height of β(px) is equal to zero. This β cannot be extended to a homomorphism M → T . This is a contradiction. Consequently, M /M is a torsion-free module. Condition (∗ ∗ M ) immediately follows from the property that the isomorphism in (∗M ) is a topological isomorphism. We only recall one property. The restriction of the homomorphism β to M can be extended to some homomorphism from M in T . This extension coincides with β , since M /M is a divisible module and T • is a reduced module. Now we assume that M /M is a divisible torsion-free module with property (∗ ∗ M ). We know that there exists a unique embedding M in M • that is the identity mapping on M (property (f)) from Sec. 21 in Part I). Consequently, we can assume that M ⊆ M ⊆ M • . The mapping in (∗M ) is injective, since the module M /M is divisible and T is a reduced module. To verify that it is surjective, we take any homomorphism β : M → T . It can be extended to a homomorphism β • : M • → T • (property (d) in Sec. 21 (Part I)). Let x ∈ M , and let y1 , . . . , yn be the elements of M whose existence is stated in (∗∗M ). We choose a positive integer k such that pk (βyi ) = 0 for all i = 1, . . . , n. Let β be the restriction of pk β • to M . Then (∗ ∗ M ) implies β (x) = 0. Therefore, β • (x) ∈ T . The restriction of β • to M is the required extension of β. Thus, the considered mapping is a bijection. It is always continuous. It follows from (∗ ∗ M ) that the inverse mapping is continuous. Consequently, property (∗M ) holds. as the sum of all submodules M in M • such that M ⊆ M and (∗M ) holds. Then We define M /M is a divisible torsion-free module, since M • /M is torsion-free. For all such M , property (∗ ∗ M ) M ). By the proved properties, (∗M ) holds. We have obtained a unique maximal submodule implies (∗ ∗ M in M • . If M 1 is one more maximal submodule, then the uniqueness property of the embedding in M • M 1 → M that coincides with (property (f) from Sec. 21 (Part I)) proves the existence of an isomorphism M the identity mapping on M . is an R-module. It follows from (1) and (2) that it remains to prove the property that M However, M • is an R-module by property (7) from Sec. 21 (Part I). Therefore, there exists an R-submodule M in M • . Then M /M is a divisible torsion-free module, since R/R generated by M is such a module (see Sec. 11 (Part I)). Why does property (∗ ∗ M ) hold? Every homomorphism β : M → T • is an R-homomorphism with coefficients in R. Since (Sec. 4 (Part I)). The elements of M are combinations of elements of M is maximal, we have M = M . ) holds, it is clear that (∗ ∗ M ) holds. Since M (∗ ∗ M • ) + M . Let us prove (3). Let α ∈ End(M ). It suffices to verify that α (M ) ⊆ M . We set M = α• (M Then M /M is a divisible torsion-free module. In addition, (∗ ∗ M ) holds. We need to only verify (∗ ∗ M ) . However, this easily follows from the application of (∗ ∗ M ) for elements of the form α• (x), where x ∈ M • • to α β , where β ∈ Hom(M , T ). Consequently, (∗M ) holds. Therefore, and α• (M ) ⊆ M . M ⊆ M
Now we formulate some sufficiently general theorem on a topological isomorphism. In this theorem, we →N ˜ such that ψ(α) and ϕ˜−1 α ˜ ϕ˜ coincide on N for keep in mind that there exists an isomorphism ϕ˜ : M every α ∈ End(M ), where ψ is some topological isomorphism End(M ) → End(N ). Theorem 34.2 (May [219]). Let M be a reduced module with nonzero torsion submodule, and let N be an arbitrary module. We assume that M/t(M ) is a divisible module. Then every topological isomorphism → N ˜ . If R is a complete of algebras End(M ) → End(N ) is induced by some module isomorphism M domain, then the assumptions on M/t(M ) can be omitted. 3332
Proof. We denote by T the torsion submodule t(M ). Let ψ : End(M ) → End(N ) be some topological isomorphism. First, we assume that R is a complete domain and M/T is a nondivisible module. Then M/T has a direct summand that is isomorphic to R (Corollary 11.8 (Part I)). Using Theorem 5.4 (Part I), we can prove that M also has a direct summand isomorphic to R. Similarly to the proof of Part (1) of Theorem 33.6, it can be proved that the corresponding summand of the module N obtained by using ψ is isomorphic to R. Thus, the modules M and N have direct summands that are isomorphic to R. Similarly to the corresponding part of the proof of Theorem 33.5, we use the method of Kaplansky to prove the →N ˜ also existence of an isomorphism ϕ : M → N inducing ψ. It is clear that the isomorphism ϕ˜ : M induces ψ. Now we assume that M/T is a divisible module and the domain R is not necessarily complete. Similarly to Theorem 33.6, it can be verified that the module N has no direct summands of the form R(p∞ ) and K, i.e., N is a reduced module. By this theorem, there exists an isomorphism ϕ : T → t(N ) such that ψ(α) and ϕ−1 αϕ coincide on t(N ) for every α ∈ End(M ). In particular, if αT = 0, then ψ(α)t(N ) = 0. Since M/T is a divisible module, Hom(M/T, M ) = 0, where we identify Hom(M/T, M ) with the set of all α ∈ End(M ) annihilating T . Therefore, Hom(N/t(N ), N ) = 0. Therefore, N/t(N ) is a divisible module, since t(N ) = 0. Consequently, M • = T • and N • = t(N )• by property (f) in Sec. 21 (Part I), ˜ ⊆ t(N )• . In addition, ϕ can be extended to an isomorphism ⊆ T • and N and we can assume that M • • • ϕ : T → t(N ) by property (d) from Sec. 21 (Part I). Endomorphisms of modules T • and t(N )• are completely determined by their action on torsion submodules. Consequently, we obtain ψ(α)• = (ϕ−1 αϕ)• = (ϕ• )−1 α• ϕ• for every α ∈ End(M ). Furthermore, the isomorphism ϕ• induces the isomorphism ψ • : End(T • ) → End(t(N )• ) extending ψ. We have ψ(α)• = ψ • (α• ). Take the isomorphism ψ −1 and use similar arguments in order to obtain the relation ψ −1 (β)• = ϕ• β • (ϕ• )−1 ) ⊆ N ˜ . By symmetry, we also have (ϕ• )−1 (N ˜) ⊆ M . As for every β ∈ End(N ). Let us assume that ϕ• (M →N ˜ , which induces ψ. a result, we obtain the isomorphism ϕ˜ : M • ˜ ) + N . Since M /M and M/T are divisible Thus, we verify that ϕ (M ) ⊆ N . We set N = ϕ• (M /T is also a divisible module. Consequently, pM +T = M . Furthermore, we have modules, M + T ) + N = pϕ• (M ) + pN + N = pN + N. N = ϕ• (pM Therefore, N /N is a divisible torsion-free module, since N • /N is a torsion-free module. We verify that ˜ , we obtain property (∗ ∗ N ) from Lemma 34.1 holds. By this lemma, (∗N ) holds. By the definition of N ˜ N ⊆ N
) ⊆ N ˜. and ϕ• (M
First, we assert that ψ(Hom(M, T )) = Hom(N, t(N )). Since ψ is a topological isomorphism, the assertion is true owing to the following property. The set Hom(M, T ) consists of all α ∈ End(M ) such that the sequence {pk α}k≥0 converges to zero in the finite topology. We also present another proof. Let α ∈ Hom(M, T ). For every y ∈ N , it follows from the continuity of ψ that there exist elements x1 , . . . , xn ∈ M such that ψ(β)y = 0 if βx1 = · · · = βxn = 0, where β ∈ End(M ). There exists a positive integer k such that pk α(xi ) = 0 for all i = 1, . . . , n. Consequently, pk ψ(α)y = 0, i.e., ψ(α)y ∈ t(N ) and ψ(α) ∈ Hom(N, t(N )). We obtain ψ Hom(M, T ) ⊆ Hom(N, t(N )). The converse inclusion is proved similarly. , property (∗ ∗ N ) holds for ϕ• (x). By property (∗ ∗ M ), We only need to verify that for every x ∈ M • we can choose elements y1 , . . . , yn ∈ M such that β (x) = 0 for every β ∈ Hom(M, T ), where β(yi ) = 0 3333
∈ N such that for all i. It follows from the topological isomorphism that there exist elements y1 , . . . , ym if β ∈ Hom(N, t(N )) and β (yj ) = 0 for all j, then β = ψ(β) for some β ∈ Hom(M, T ), where β(yi ) = 0 for all i. However, β • (x) = 0. Consequently,
(β )• (ϕ• (x)) = ψ(β)• (ϕ• (x)) = (ϕ• ψ(β)• )(x) = (β • ϕ• )(x) = ϕ• (β • (x)) = 0. Therefore, (∗ ∗ N ) holds and the proof is completed. = M . As usual, M 1 denotes the first Ulm submodule We need one condition that implies the relation M of the module M . Lemma 34.3. Let M be a reduced module with unnbounded torsion submodule over a complete domain = M. R. If M 1 is a cotorsion module, then M \ (M 1 + M ). Let y1 , . . . , yn ∈ M be the elements Proof. We assume that there exists an element x ∈ M ) of Lemma 34.1. We set from property (∗ ∗ M . 1 + Ry1 + · · · + Ryn ⊆ M A=M by Proposition 28.4 (2). It is clear that Since R is a complete domain, A is a nice submodule in M 1 /A. There exists a = ∅. Consequently, the residue class x + A has finite height, say, k in M (x + A) ∩ M composition of the mappings /A) → t(M ) →M /A → (M /A)/pk+1 (M β:M ). Consequently, M =M 1 + M . such that β(yi ) = 0 for every i, but β(x) = 0. This contradicts (∗ ∗ M /M is a torsion-free module, we have M 1 ∩ M = M 1 . Now Since M 1 /(M /M = (M 1 + M )/M ∼ 1 ∩ M ) = M 1 /M 1 . M =M 1 = M 1 , 1 /M 1 is a divisible torsion-free module or the module is equal to zero. If M Consequently, M 1 , which is impossible, since M 1 is a reduced then the cotorsion module M 1 is a direct summand in M 1 1 = M. = M and M module. Consequently, M Theorem 34.4 (May [219]). Let R be a complete domain, M be a reduced module, and ψ : End(M ) → End(N ) be a topological isomorphism. (1) If M 1 is a cotorsion module, then ψ is induced by some embedding from N in M . (2) If M 1 is a cotorsion module, then every topological automorphism of the algebra End(M ) is an inner automorphism. (3) If M 1 is the direct sum of a bounded module and a torsion-free module of finite rank, then ψ is induced by some isomorphism from M onto N . In particular, this is true provided M 1 = 0. Proof. If t(M ) is a bounded module, then M = t(M ) ⊕ X, where X = 0 or X is some torsion-free module. If X = 0, then ψ is induced by some isomorphism from M onto N by Theorem 33.6. If X = 0, then the module M has a direct summand isomorphic to R. In this case, N has a similar summand and ψ is induced by some isomorphism from M onto N (see the beginning of the proof of Theorem 34.2). Consequently, we can assume that t(M ) is a unbounded module. By Theorem 34.2, ψ is induced by some →N ˜ . However, it follows from Lemma 34.3 that M = M . Consequently, ϕ˜−1 is an isomorphism ϕ˜ : M |N injection N → M that induces ψ in the sense that ψ(α) and ϕ˜−1 αϕ˜ coincide on N for every α ∈ End(M ). In (2), we have ϕ˜ : M → M and ϕ˜ induces ψ. In (3) M 1 = t(M ) ⊕ F , where t(M ) is a bounded module and F is a free module of finite rank. In particular, M 1 is a complete module; therefore, M 1 is a cotorsion module of finite rank. By (1), there exists an embedding N → M . Therefore, N 1 is a cotorsion module, since it is isomorphic to some = M, N ˜ = N , and ϕ˜ : M → N induces ψ. submodule in M 1 . Therefore, M 3334
In general, Parts (2) and (3) are not true without the corresponding condition for the first Ulm submodule. We present examples that justify this. We need the notion of small homomorphism (see remarks to Chapter 4 (Part I)). Let M and N be two primary modules. A homomorphism ϕ : M → N is said to be small if for every positive integer e, there exists a positive integer n with ϕ((pn M )[pe ]) = 0. All small homomorphisms from M in N form an R-module Small(M, N ). All small endomorphisms of the module M form an ideal Small(M ) of the algebra End(M ). Let G be a module with torsion submodule T . Then T and T 1 are fully invariant submodules in G. Consequently, every endomorphism β of the module G induces the endomorphism β¯ of the factor module ¯ + T 1 ) = β(x) + T 1 , x ∈ T . The endomorphism β is said to be small with respect T /T 1 by the relation β(x 1 ¯ to the module T if β is a small endomorphism of the module T /T 1 . All small endomorphisms of the module G with respect to T 1 form an ideal in End(G). An automorphism of a ring or an algebra that is not an inner automorphism is called an outer outer automorphism. Proposition 34.5 (May [219]). Let R be a complete domain. There exists a mixed module M of rank 1 such that the algebra End(M ) has a topological outer automorphism. Proof. We use a known result on the existence of a primary module with a given Ulm sequence (Fuchs [90, Theorem 76.1]). We also apply theorems from [48] and [58] on a split realization to endomorphism rings of primary modules (also see the remarks at the end of Chapter 4 (Part I)). It follows from these results that we can choose primary modules T1 and T2 such that Ti1 is an unbounded direct sum of cyclic modules, End(Ti /Ti1 ) = R ⊕ Small(Ti /Ti1 ) (direct sum of R-modules), and Hom(Ti /Ti1 , Tj /Tj1 ) consists of small homomorphisms for i, j = 1, 2 and i = j. We set T = T1 ⊕ T2 . We have T • = T1• ⊕ T2• , and denote by εi the projection T • → Ti• , i = 1, 2. Let us prove that End(T • ) = Rε1 ⊕ Rε2 ⊕ I (direct sum of R-modules), where I is the ideal of small endomorphisms of the module T • with respect to the module T 1 . First, we verify that End(T1 ) = R ⊕ I1 , where I1 is the ideal of small endomorphisms of the module T1 with respect to the module T11 . It is clear that R∩I1 = 0. Let α ∈ End(T1 ), and let α ¯ be an ¯ where r ∈ R and β¯ ∈ Small(T1 /T 1 ). ¯ = r + β, endomorphism of the module T1 /T11 induced by α. Then α 1 Since α = r + (α − r), we have α − r = α ¯ − r¯ = α ¯ − r = β¯ and α − r ∈ I1 . We obtain End(T1 ) = R ⊕ I1 . Similarly, we have End(T2 ) = R ⊕ I2 , where I2 is the ideal of small endomorphisms of the module T2 with respect to the module T21 . Furthermore, we have End(T ) = End(T1 ) ⊕ End(T2 ) ⊕ Hom(T1 , T2 ) ⊕ Hom(T2 , T1 ) = Rε1 ⊕ Rε2 ⊕ I. Now note that the ring End(T • ) can be identified with the ring End(T ) via the restriction mapping (see property (d) from Sec. 21 (Part I)). By using Ext, the submodule Ti1• (respectively, T 1• ) can be naturally embedded in Ti• (respectively, T • ). Since Ti1 is an unbounded direct sum of cyclic modules, (Ti1• )1 is a torsion-free module by Proposition 21.5 (Part I). We choose nonzero elements xi ∈ (Ti1• )1 , i = 1, 2. We set x = x1 +x2 . Let M be a pure submodule in T • such that T ⊆ M and M/T = K(x + T ), where K is the field of fractions of the domain R. (The divisible torsion-free module T • /T is a K-space.) Let β ∈ I. It is easy to verify that βT 1 = 0 (see Exercise 1). Consequently, βT 1• = 0 and βx = 0. Therefore, we have R ⊕ I ⊆ End(M ). If α ∈ End(M ) (say, α = r1 ε1 + r2 ε2 + β, where r1 , r2 ∈ R and β ∈ I), then α(x) = r1 x1 + r2 x2 . Then α(x) ∈ M iff r1 = r = r2 . Consequently, α = r + β ∈ R ⊕ I and End(M ) = R ⊕ I. In particular, End(M ) ⊂ End(T • ). We choose some invertible element u ∈ R that is not equal to 1 and set γ = ε1 + uε2 . Here γ is an automorphism of the module T • (indeed, γ −1 = ε1 + u−1 ε2 ). By the relation ψ(α) = γ −1 αγ with α ∈ End(T • ), an automorphism of the algebra End(T • ) is defined. Since I is an ideal, the restriction of ψ is an automorphism of the algebra End(M ). We assume that ψ is an inner automorphism of the 3335
algebra End(M ). Then there exists an automorphism ϕ of the module M such that ψ(α) = ϕ−1 αϕ for every α ∈ End(M ). Consequently, we have α(ϕγ −1 ) = (ϕγ −1 )α for every α ∈ End(M ). In particular, ϕγ −1 commutes with any endomorphism that is small with respect to the module T 1 . In the last part of the proof of Theorem 33.1, all of the considered endomorphisms, except for γ, can be chosen small with respect to the module T 1 . As a result, we obtain that ϕγ −1 is contained in the center of the ring End(T ), i.e., ϕγ −1 is an invertible element from R, whence γ ∈ End(M ). However, γ(x) = x1 + ux2 ∈ / M ; this is a contradiction. Consequently, ψ is an outer automorphism. For the proof of the property that ψ is continuous in the finite topology, we take an element y ∈ M . It suffices to verify that there exists an element y1 ∈ M such that if α ∈ End(M ) and α(y1 ) = α(x) = 0, then ψ(α)(y) = 0. Noting that γ is an automorphism, we have α(γ −1 y) = 0. For some k ≥ 0 and a ∈ R, we have pk y = a(x1 + x2 ). By the choice of x2 , we have x2 = pk x3 , where x3 ∈ T 1• . Then pk γ −1 (y) = a(x1 + u−1 x2 ) = pk y + pk a(u−1 − 1)x3 . We set y1 = γ −1 (y) − a(u−1 − 1)x3 and note that y1 ∈ M , since y1 ∈ y + T . Let α = r + β ∈ End(M ) and let α(y1 ) = α(x) = 0. Since β(x) = 0, we have r = 0; therefore, α = β. Since β(x3 ) = 0, we have α(γ −1 y) = 0, which is what we were required to prove. By symmetry, ψ −1 is continuous and ψ is a topological automorphism. In Part (3) of Theorem 34.4, the condition for the submodule M 1 is the best (in some sense) possibility for such theorems of a topological isomorphism. Using constructions similar to those used in Proposition 34.5, May [219] has obtained the following results. Let A be a reduced module over a complete domain R such that t(A)1 = 0 and A is not a direct sum of a bounded module and a torsion-free module of finite rank. Then there exists a reduced module M such that M 1 ∼ = A and M does not satisfy the theorem of a topological isomorphism in the weak sense. If A is a primary module, then we can take a mixed module of rank 1 as M . In addition, there exists a mixed module N of rank 1 such that the algebras End(M ) and End(N ) are topologically isomorphic to one another but the modules M and N are not isomorphic. The following exercises are taken from the paper of May [219]. Exercise 1. Let M be a module and let T = t(M ). If β is a small endomorphism of the module M with respect to T 1 , then β(T 1 ) ⊆ T 2 . Let S be some R-algebra. If, as a sub-basis of neighborhoods of zero, we take the set of right annihilators of elements of the right ideal in S generated by primitive idempotents of finite additive order in S, then we obtain a topology on S called the intrinsic topology. Exercise 2. Prove that for a primary module T that not a direct sum of a bounded module and a nontrivial divisible module, the intrinsic topology on End(T ) coincides with the finite topology. (See Proposition 16.2 (Part I); it is also useful to compare the intrinsic topology with two finite topologies defined before Theorem 18.4 (Part I) and Proposition 18.6 (Part I).) Exercise 3. Let M be a module with torsion submodule T . We assume that the ring End(M ) is Hausdorff in the intrinsic topology. (1) The restriction mapping f : End(M ) → End(T ) is injective. (2) End(T )/f End(M ) is a torsion-free R-module. (3) The intrinsic topology on End(M ) is induced by the finite topology on End(T ). Exercise 4. The endomorphism ring of the module M from Proposition 34.5 is complete in the intrinsic topology and End(M ) is not isomorphic to End(P ) for every primary module P . 35.
Modules over Completions
be the p-adic completion of R. We show Let R be an incomplete discrete-valuation domain and let R that every reduced R-module can be embedded in a minimal (in a certain sense) reduced R-module. In 3336
general, modules over complete domains have more simple structures. Therefore, such an embedding can be useful in the study of modules over incomplete domains. At the end of this section, we briefly present several results on the isomorphism problem for endomorphism rings of mixed modules; some of the results use embeddings of R-modules in R-modules. Other results on this topic are contained in the next section. In the remaining two sections of the chapter, arbitrary (i.e., not necessarily topological) isomorphisms are considered. It is clear that nontopological isomorphisms more weakly reflect the structure of original modules. In this connection, it is interesting that every isomorphism between endomorphism algebras of two Warfield modules is a topological isomorphism (Theorem 36.4). In this and subsequent sections, we denote by RX the submodule generated by a subset X of some module. In Chapter 6, we have agreed that for a mixed module M , the rank of the factor module M/t(M ) is called the rank of M . If this factor module is a divisible module, then it is a K-space, and its dimension is equal to the rank of the module M . Our constructions are based on the properties of cotorsion hulls and tensor products. Let M be a reduced R-module. Similarly to the previous section, we consider M as a submodule of the cotorsion hull M • such that t(M • ) = t(M ) and M • /M is a divisible torsion-free module. By property (7) from Consequently, we can take the R-submodule in M • generated by Sec. 21 (Part I), M • is an R-module. the submodule M . We denote it by RM ; it is called the R-hull of the module M . By construction, RM ⊗ M is an R-module is a reduced module. It was mentioned in Sec. 13 (Part I) that the tensor product R (all tensor products are considered over the ring R). The R-hull of the module M can be constructed by using of this tensor product. For this purpose, we consider the exact sequence of R-modules ⊗ M → R/R 0→R⊗M →R ⊗ M → 0. The module R ⊗ M can be identified with M under the correspondence r ⊗ m → rm (r ∈ R, m ∈ M ), ⊗ M . Furthermore, R/R and M can be identified with the image in R ⊗ M is a divisible torsion-free module, since R/R has an analogous structure. We have R ⊗ M = D ⊕ A, where D is the largest divisible ⊗ M and A is a reduced R-module R A such that M ⊆ A (see the remark submodule of the R-module after Theorem 6.1 in Part I). is isomorphic to the factor module of the R-module ⊗ M with respect Lemma 35.1. The R-hull RM R ⊗ M. to the largest divisible submodule of R × M → RM , (r, m) → rm induces an epimorphism of R-modules g : Proof. A balanced mapping R R ⊗ M → RM , r ⊗ m → rm. In addition, gD = 0, since RM is a reduced module. Let H be the submodule in R ⊗ M generated by all elements of the form ri ⊗ mi − 1 ⊗ ri mi , where ri ∈ R, +R = R that H is divisible. If mi ∈ M , and ri mi ∈ M . It easily follows from the relation pR ri ⊗ mi ∈ Ker(g), then ri ⊗ mi = ri ⊗ mi − 1 ⊗ ri mi ∈ H ⊆ D; ∼ ⊗ M )/D. therefore, Ker(g) = D. Consequently, RM = (R In the second proof, the above exact sequence of tensor products is used. Since A/M is a divisible torsion-free module, it follows from property (f) of Sec. 21 (Part I) that M • = A• , and we can assume that M ⊆ A ⊆ M • . Let h be the restriction of g to A and let h• : M • → M • be an extension of h. Since (see assertion h coincides on M with the identity mapping, h• is the identity automorphism and A = RM (d) in Sec. 21 (Part I)). We present several simple properties of R-hulls. They follow from the definition of the R-hull and Lemma 35.1. ) = t(M ), and M is an isotype submodule in RM (1) RM/M is a divisible torsion-free module, t(RM (such submodules were introduced in Sec. 27). 3337
) = t(M ). Since the module Since M • /M is torsion-free, we have that RM/M is torsion-free and t(RM R/R is divisible, the module RM/M is divisible (the proof of Lemma 34.1 has a similar point). (2) The module RM can be characterized as a reduced R-module such that RM contains M as an is generated by M as an R-module, R-module, RM and RM/M is a divisible torsion-free module. does not exceed the rank of the R-module M . (3) The rank of the R-module RM If {ai }i∈I is a maximal linearly independent system of elements of infinite order of the module M , then ) {ai + t(M )}i∈I is a basis of the K-space K(M/t(M )) (see Sec. 4 (Part I)). Every element of RM/t(M can be linearly expressed through this basis; this yields the required result. (4) Let Mi (i ∈ I) be the set of reduced R-modules. Then i. RM R Mi ∼ (1) = i∈I
i∈I
Since Ext(K/R, −) commutes with finite direct sums of modules, we assume that • • Mi ⊆ Mi . i∈I
i∈I
Under such an approach, the isomorphism in (1) becomes an equality. We can also use Lemma 35.1. The canonical isomorphism ⊗ ⊗ Mi ) R Mi ∼ (R = i∈I
i∈I
implies the required isomorphism of factor modules with respect to divisible submodules. (5) We obtain a covariant functor F from the category of reduced R-modules into the category of and letting F (f ) be the induced homomorphism RM → RN reduced R-modules by setting F (M ) = RM for any two R-modules M and N and any homomorphism f : M → N . Here, F (f ) can also be directly f • is an R-module homomorphism M • → N • . Then F (f ) is defined. Since M • and N • are R-modules, . We can also define F (f ) using the induced homomorphism 1 ⊗ f of tensor the restriction of f • to RM products. ) (i.e., the R-endomorphisms of the module RM (6) End(M ) is a subalgebra of the algebra End(RM coincide with the R-endomorphisms of RM ). For every endomorphism α of the module M , we take the corresponding F (α) as in (5). In other words, . we extend α to an endomorphism α• of the module M • ; then we take the restriction of α• to RM Isomorphic modules have isomorphic R-hulls. For Warfield modules, the converse assertion holds. , then M ∼ ∼ Proposition 35.2 (Files [71]). If M and N are two Warfield modules and RM = N. = RN Proof. Since every primary module M is an R-module, we have RM = M . We can assume that M and N are not primary modules. Let ϕ : RM → RN be some isomorphism. There exists a primary module T such that M ⊕ T = Mi and N ⊕ T = NJ , where all modules Mi and Nj have rank 1 i∈I
j∈J
(Corollary 32.3 and Theorem 30.3). For every i ∈ I, we choose an element xi ∈ Mi ∩ M of infinite order. . It is necessary to consider M and RM Then X = {xi | i ∈ I} is a decomposition basis for the R-modules the property that, by (4) and (3), we have i, ⊕T = RM RM i∈I
i is a module of rank 1. Similarly, we take an element of infinite order yj in every Nj ∩ N where every RM . Since ϕX and Y are two decomposition and obtain a decomposition basis Y = {yj | j ∈ J} for N and RN 3338
, there exist subordinates X = {x | i ∈ I} in X and Y = {y | j ∈ J} in Y , respectively, bases for RN i j and the bijection τ : I → J such that k k hRM (p ϕ(xi )) = hRN (p yτ (i) )
for all i ∈ I and k ≥ 0 (see the paragraph after the proof of Corollary 31.2). Since the submodules M and N are isotype, we have hM (pk xi ) = hN (pk ϕ(xi )) = hN (pk yτ (i) ) for all i and k. We obtain an isomorphism χ : RX → RY that preserves heights in M and N if we set χ(xi ) = yτ (i) for every i ∈ I. In addition, we have )∼ ) = t(N ) t(M ) = t(RM = t(RN and M ∼ = N by Corollary 32.7. Every R-module M has the largest R-submodule. We denote it by C(M ). This submodule plays some role in the next section. Lemma 35.3. If M is a reduced module containing a nice basis, then C(M ) = t(M ). Proof. Let X be a nice basis for M . The torsion submodule t(M ) is an R-module; therefore, it is contained ⊆ M. in C(M ). If C(M ) = t(M ), then we can choose a nonzero element x ∈ C(M ) ∩ RX. Then Rx Therefore, ∼ R/R ⊆ M/Rx, = Rx/Rx and M/Rx is an irreduced module. Since M/RX ∼ = (M/Rx)/(RX/Rx) and RX/Rx is a reduced module, M/RX is an irreduced module. However, this is impossible, since RX is a nice submodule. Consequently, C(M ) = t(M ). As a rule, the Kaplansky method is not applicable to endomorphism rings of mixed modules. A formal obstruction is the absence of endomorphisms that map from elements of infinite order onto cyclic direct summands of the module. Practically all methods of solution of the isomorphism problem for endomorphism rings of mixed modules are based on the use of cotorsion hulls. This was partially shown in Sec. 34. We can assume that every reduced module is standardly embedded in the cotorsion hull of the module. Now we assume that the endomorphism algebras of reduced mixed modules M and N are isomorphic to one another. In this case, it can usually be proved that the torsion submodules t(M ) and t(N ) are isomorphic to one another (see Theorem 36.4). Furthermore, the general case can be frequently reduced to the case where M/t(M ) and N/t(N ) are divisible modules. This is one of the common situations. For example, such a reduction is always possible if R is a complete domain. Then M • = T • = N • , where T = t(M ), and we can assume that the modules M and N are contained in the cotorsion hull T • . Furthermore, the algebras End(M ) and End(N ) can be considered as subalgebras in End(T • ) if we identify every endomorphism α ∈ End(M ) with α• ∈ End(T • ) and do the same with End(N ). May [215] has proved that End(M ) = End(N ) in this case. In addition, the original isomorphism ψ of endomorphism algebras is induced by an isomorphism from M onto N iff there exists an invertible with uM = N . If R is a complete domain, then ψ is induced by an isomorphism from M element u ∈ R onto N iff M = N . Therefore, the problem is to find the element u. If ψ is not induced by any isomorphism from M onto N (for example, this case is possible for Warfield modules, as in Theorem 36.4), then we need to continue the search for conditions implying the existence of an isomorphism between M and N . Almost all existing results on the isomorphism problem for mixed modules are related to modules with totally projective torsion submodule, Warfield modules, or some larger classes of modules related to these two classes. The following theorem of May and Toubassi [221] can be considered as the first result devoted to this topic. 3339
Theorem 35.4. Let M be a mixed module of rank 1 with totally projective torsion submodule. If N is a module of rank 1, then every isomorphism End(M ) → End(N ) is induced by some isomorphism M → N . In [217], May found sufficiently general conditions that imply that an isomorphism End(M ) ∼ = End(N ) ∼ is induced by an isomorphism of R-hulls RM RN , where M is some mixed module and N is a = general module. He obtained some interesting applications to modules M with totally projective torsion submodule and Warfield modules M . If R is a complete domain, then we obtain the isomorphism of the modules M and N . For modules over a complete domain, May proved a stronger theorem in [215]. Theorem 35.5. Let M be a reduced mixed module over a complete domain R. We assume that every submodule G in M such that M/G is a primary module contains a nice submodule A such that M/A is a totally projective module. Then every isomorphism from End(M ) onto End(N ) is induced by some isomorphism from M onto N . May has proved that the assertion of the theorem holds if the module M satisfies one of the following conditions: (1) the rank of M does not exceed X0 and t(M ) is a totally projective module; (2) M is a Warfield module. May presented examples that imply that Theorem 35.5 does not hold provided that either t(M ) is not totally projective, or the rank of M exceeds X0 , or M is not reduced. In each of these cases, the corresponding endomorphism algebras have outer automorphisms. We present one of the examples in greater detail. The example demonstrates obstructions to the search for isomorphism theorems for mixed modules. Example 35.6. There exist a primary module T without elements of infinite height and a continual set of mixed modules Mi of rank 1 such that t(Mi ) = T , Mi /T is a divisible module, End(Mi ) = R ⊕ Small(T ), and Hom(Mi , Mj ) = Small(T ) for all distinct i, j < 2X0 . Each of the algebras End(Mi ) has an outer automorphism. In the following exercises, M is a reduced module. ) holds. The σ M ) ⊆ pσ (RM Exercise 1 (May [217]). For every ordinal number σ, the inclusion R(p σ σ ) holds if either σ is finite, or M has rank 1, or M is a Warfield R-module. M ) = p (RM relation R(p ⊗ M is equal to the rank of the module M (see property (3)). Exercise 2. The rank of the R-module R Exercise 3. Prove that C(M ) is a fully invariant submodule in M and M/C(M ) is a torsion-free module. 36.
Endomorphisms of Warfield Modules
We prove that a Warfield module is determined by the endomorphism algebra (in the weak sense) of the module in the class of such modules. We base our analysis on the paper of Files [71]. We widely use R-hulls introduced in the previous section. As earlier, our constructions are based on properties of cotorsion hulls. (The sense of these constructions is presented at the end of Sec. 35.) Let M be a reduced module such that M/t(M ) is a divisible module. We set T = t(M ). By property (f) from Sec. 21 (Part I), T • = M • . We recall that T • is an adjusted cotorsion module (Proposition 21.3 (Part I)). Every endomorphism α of the module M has a unique extension to an endomorphism α• of the module T • (property (d) in Sec. 21 (Part I)). In such a manner, End(M ) can be embedded in End(T • ). Similar arguments are true for every pure submodule N in T • containing T . Therefore, we consider End(M ) and End(N ) as subalgebras of the same algebra End(T • ). Under such conditions, we have End(M ) = End(N ) iff α• (N ) ⊆ N for all α ∈ End(M ) and β • (M ) ⊆ M for all β ∈ End(N ). We note that End(T ) = End(T • ), since the submodule T is fully invariant. The Ulm submodules M σ and the factors Mσ were defined in Sec. 28. 3340
Lemma 36.1. Let T be a reduced primary module. We assume that M and N are pure submodules in T • that contain T and have nice decomposition bases. If End(M ) = End(N ), then Hom(M, T ) = Hom(N, T ). Proof. By Lemma 35.3, C(M ) = T = C(N ). It suffices to prove that Hom(M, C(M )) = Hom(N, C(N )). Consequently, We take α ∈ Hom(M, C(M )). Then rα(M ) ⊆ M . Consequently, rα• (N ) ⊆ N for all r ∈ R. the R-submodule generated by α• (N ) in N is contained in N . Therefore, α• (N ) ⊆ C(N ) and α ∈ Hom(N, C(N )). We obtain one inclusion. The other inclusion follows by symmetry. A delicate interrelation exists between the heights of elements of infinite order of the cotorsion hull T • and the length l(T ) of a reduced primary module T (the length is defined in Sec. 28). In particular, to construct certain endomorphisms, we need to know that cotorsion hulls of some primary modules contain elements of infinite order and sufficiently large height. Papers of May [215, 217] contain information on this topic. We need one result from [215]. In the proof of Lemma 36.2, the notion of the inverse limit is considered; the notion is presented in [89]. (The dual construction of the direct limit was introduced in Sec. 29.) Lemma 36.2. Let T be a nonbounded primary module such that T ⊆ T¯, where T¯ is a totally projective module of length < l(T ) + ω. Let σ be the least ordinal number such that T σ is a bounded module. If τ < σ, then the rank of the module (T • )τ is no less than 2X0 . The rank of the module (T • )σ is at least 2X0 except for the case where σ is a limit ordinal number of confinality > ω; in this case (T • )σ is a primary module. Proof. It is clear that σ is also the least ordinal number such that T¯σ is a bounded module. We choose n with pn (T¯σ ) = 0. If τ < σ, then Tτ is an unbounded module. It follows from [89, Theorem 56.7] that (T • )τ has a summand that is isomorphic to the p-adic completion Tτ of the module Tτ (see also Proposition 21.5 (Part I)). Therefore, the rank of the module (T • )τ is at least 2X0 . We assume that σ is a nonlimit ordinal number. Let σ = τ + 1 for some τ . It follows from the above properties that the rank of the module (T • )σ is no less than the rank of the maximal divisible torsion submodule of the module Tτ /Tτ . The embedding T → T¯ induces the homomorphism Tτ → T¯τ such that the orders of elements of the kernel are bounded by the element pn . The image G of this homomorphism should be an unbounded module, since Tτ is a unbounded module. Furthermore, T¯τ is the direct sum of cyclic modules, since the totally projective module T¯ is simply presented (see the end of Sec. 30 and property (f)). We have an epimorphism Tτ → G such that the kernel of it is bounded by the element pn . and T¯τ /Tτ → G/G. The rank of the maximal The same also holds for the induced mappings Tτ → G X 0 divisible torsion submodule of the module G/G is at least 2 . Consequently, the same holds for Tτ /Tτ . The assertion has been proved for σ = τ + 1. Now we assume that σ is a limit ordinal number. We take inverse spectra of modules {T /T τ , τ < σ; πρτ } ¯ρτ }, where πρτ (ρ ≤ τ ) is the canonical homomorphism T /T τ → T /T ρ and π ¯ρτ has and {T¯/T¯τ , τ < σ; π ¯ be the limits of these spectra. As earlier, it is sufficient to prove the a similar sense. Let L and L corresponding result on the rank of the maximal divisible torsion submodule of the module L/(T /T σ ). Let ϕ : T → T¯/T¯σ be the restriction of the canonical homomorphism. Then the set of orders of elements of the kernel ϕ is bounded by the element pn . Since T¯/T¯σ is a totally projective module and the Ulm length of the module is a limit ordinal number, we have T¯/T¯σ = Aτ , where each of the modules τ ω (see the remark after the proof of Lemma 36.2). We have l(t(G)) = l(t(G)• ) = λ, whence hG (pk y + A• ) < λ for all k ≥ 0. Let μ be the supremum of the heights hG (pk y + A• ) for all k ≥ 0. Then μ + ω < σ ≤ l(T • ) by the confinality of σ. We have proved the existence of μ. By the remark after Lemma 36.2, we can choose an element z ∈ pμ T • of infinite order. Since G is a module of rank 1 with totally projective module t(G), we have that R(y + A• ) is a nice submodule in G and G/R(y + A• ) is a totally projective module (see the paragraph before Corollary 32.9). The mapping ry + A• → rz (r ∈ R) from R(y + A• ) into T • does not decrease the heights with respect to G and T • . Consequently, the mapping can be extended to a homomorphism G → T • (Corollary 32.8). Furthermore, we take the composition of the canonical mapping M → M/A ⊆ G with this homomorphism and extend the composition to an endomorphism α of the module T • . We obtain αy = z and αA = 0. Therefore α ∈ Hom(M, T ) = Hom(N, T ) and αy ∈ / T . Consequently, y ∈ / N. and the converse inclusion follows from We show that M ⊆ RN . This implies the inclusion RM ⊆ RN . We take an element x ∈ X. We have A = Rx ⊕ C and the symmetry. It suffices to prove that X ⊆ RN • • A = Rx ⊕ C . Since X is a nice decomposition basis for M and M/A is a totally projective module, the projection A → Rx extends to an endomorphism π of the module M (Corollary 31.11). Since πM T , we have π • N T , say, π • (b) ∈ / T for some b ∈ N . By the properties proved, N ⊆ (A• )∗ and we can 0 = r ∈ R, and c ∈ C • . assume that b ∈ A• . Therefore, b = vrx + c, where v is an invertible element in R, • • We have π (N ) ⊆ N , since π ∈ End(M ). Consequently, vrx = π (vrx + c) = π • (b) ∈ N . Therefore, and X ⊆ RN , since X is pure. rx = v −1 π • (b) ∈ RN Theorem 36.4 (Files [71]). We assume that M is a Warfield module and N = N1 ⊕ D, where N1 is a Warfield module and D is a divisible module. If ψ is some isomorphism from End(M ) onto End(N ), then ψ is a topological isomorphism and M ∼ = N. Proof. If D is a nonzero module, then it contains a summand of the form R(p∞ ) or K. Similar to Theorem 33.6, we can take the inverse isomorphism ψ −1 and obtain that M has a summand that is The first two cases are impossible. In the remaining isomorphic to one of the modules R(p∞ ), K, or R. ⊆ C(M ); this is impossible by Lemma 35.3. Thus, N = N1 is a reduced module. case, we obtain R First, we assume that M/t(M ) is not a divisible module. We choose a primary module P such that M ⊕ P is equal to the direct sum of modules Mi , i ∈ I, of rank 1 (Corollary 32.3 and Theorem 30.3). There exists a subscript i such that Mi /t(Mi ) is a reduced module; therefore, the module is isomorphic to R. Consequently, M/t(M ) has R as a direct summand. Then M also has a similar summand (see Theorem 34.2). Similarly to Theorem 33.5, we can verify that N has a summand R and ψ is induced by an isomorphism from M onto N . In particular, ψ is a topological isomorphism. We assume that M/t(M ) is a divisible module. If M is a primary module, then End(M ) is an R algebra; therefore, End(N ) is an R-algebra. Therefore, N is an R-module and N = C(N ). It follows from Lemma 35.3 that N is a primary module. This can also be proved by using the standard methods of cyclic summands (for example, see the proof of Theorem 33.6). It remains to refer to Theorem 33.2. We can assume that M is a mixed module. It follows from the properties proved that N/t(N ) is a divisible module. We set T = t(N ). The Kaplansky method (which is used in proving Theorem 33.6 (1)) can also be used in this case, since M and N are reduced modules. As a result, we can construct an isomorphism ϕ : t(M ) → T such that ψ(α) and ϕ−1 αϕ coincide on T for all α ∈ End(M ). Let M = ϕ• (M ) ⊆ T • . Then End(M ) = End(N ). If β ∈ End(M ), then β = ϕ• β(ϕ• )−1 ∈ End(M ). Consequently, ψ(β ) ∈ End(N ). However, ψ(β ) and (ϕ• )−1 β ϕ• = β coincide on T . Consequently, β • = ψ(β )• and β • ∈ End(N ). 3343
We obtain End(M ) ⊆ End(N ). The converse inclusion can be proved by using similar arguments. . By Proposition 35.2, M = N , whence M ∼ = RN Proposition 36.3 implies RM = N. Finally, why is ψ a topological isomorphism? First, we verify that the identification End(M ) = End(N ) , we have y = r1 x1 +· · ·+rn xn , ri ∈ R and xi ∈ M . It is clear = RN is topological. Let y ∈ N . Since RM • • that if α ∈ End(M ) and αxi = 0 for all i, then α (y) = 0 (α is an R-module endomorphism). It can be asserted that neighborhoods of zero remain neighborhoods of zero under the identification. This implies a topological nature of the identification. The isomorphism ψ is topological, since it is the composition of the identification with the isomorphism induced by ϕ• . (The induced isomorphism is always a topological isomorphism; see Sec. 33.) Corollary 36.5. If M is a Warfield module, then every automorphism of the algebra End(M ) is a topological automorphism. Files [71] has constructed two modules M and N such that the isomorphism ψ from Theorem 36.4 is not induced by any isomorphism from M onto N . The corresponding domain R must be incomplete. In the case of the complete domain R, it follows from Theorem 35.5 that ψ is always induced by an isomorphism from M onto N , where we do not impose any restrictions on N . In the example of Files, the algebra End(M ) has an outer automorphism. In [72], Files has found when the endomorphism ring of a p-local Warfield group (i.e., a Zp -Warfield module) has outer automorphisms. All exercises are taken from the papers of Files [70, 73]. An exact module is defined in remarks at the end of the chapter. Exercise 1. Let M be a reduced R-module that is not primary. Prove that the following conditions are equivalent: (1) M is an exact module; ⊗ M with respect to the maximal divisible submodule is an (2) the factor module of the module R R-Warfield module of rank 1; (3) M is contained in the R-Warfield module M of rank 1 and M /M is a torsion-free module. Exercise 2. (a) A Warfield module of rank 1 is an exact module. (b) A reduced, torsion-free R-module M is an exact module iff M is isomorphic to some pure R submodule in R. (c) A domain R is not complete iff there exists an exact R-module of rank > 1. )=R EndR (M ). Exercise 3. If M is a sharp R-module, then EndR (RM Exercise 4. Prove that a reduced module M has a stable element (the definition of it is presented below) in the following cases: (1) the rank of M is equal to 1; (2) M is a Warfield module and the set of height sequences of elements of some decomposition basis for M contains the least element; (3) M = A ⊕ B, where A has a stable element and B/ Hom(A, B)A is a primary module. Remarks. Additional information on the isomorphism problem for endomorphism rings is contained before the exercises in Secs. 34–36. We present several remarks. Files [73] introduced the notion of a stable element of a mixed module. The properties of such an element resemble properties of an element from the direct summand that is isomorphic to R. Precisely, an element x of the module M is said to be stable if the following conditions hold: (1) there exists an endomorphism ρ ∈ End(M ) such that ρx = x and ρM/Rx is a primary module; (2) M/ End(M )x is a primary module. Files has proved the following theorem. We assume that M and N are reduced modules and t(M ) is a totally projective module. If each of the modules has a stable element, then every isomorphism 3344
End(M ) → End(N ) is induced by some isomorphism M → N . Some applications are obtained (e.g., to Warfield modules). The paper of May [218] is interesting. It contains the following isomorphism theorem in the weak sense for a sufficiently large class of mixed modules that are not related to totally projective modules or Warfield modules. Let M be a module over a complete domain such that M/M 1 is a mixed module of countable rank. Then the isomorphism End(M ) ∼ = End(N ) implies the isomorphism M ∼ = N. The conditions of the theorem hold for a module M of countable rank that contains an element such that all coordinates of the height sequence of it are finite. An isomorphism of endomorphism algebras that is not induced by a module isomorphism, i.e., in the general case, the module M does not satisfy the isomorphism theorem in the strong sense was constructed. In [70], a reduced module M is said to be of M is a Warfield module of rank 1. It is proved that, for an exact module M , exact if the R-hull RM all automorphisms of the algebra End(M ) are inner automorphisms. There is a large bibliography on isomorphisms of endomorphism rings of modules over various rings (see reviews of Mikhalev [227] and Markov, Mikhalev, Skornyakov, and Tuganbaev [211]). The papers of Wolfson [311, 312] and Franzsen and Schultz [86] are devoted to endomorphism rings of modules that are close to free. In the last paper, the authors analyze the situation where an isomorphism End(M ) → End(N ) is not induced by any semilinear isomorphism M → N but there exists a semilinear isomorphism M → N . The interrelation between this situation and the existence of inner or outer automorphisms of the ring End(M ) is discussed. There exist studies related to isomorphisms of endomorphism groups and endomorphism semigroups of modules (in this connection, see the mentioned reviews); we mean the additive group and the multiplicative semigroup of the endomorphism ring, respectively. Let S1 and S2 be two rings. An additive isomorphism f : S1 → S2 is called an antiisomorphism if f (xy) = f (y)f (x) for all x, y ∈ S1 . An anti-isomorphism of a ring onto itself is called an antiautomorphism. Wolfson [313] considered the problem on conditions under which an antiisomorphism of endomorphism rings of two locally free modules is induced by a semilinear antiisomorphism of these modules. For example, if endomorphism rings of two reduced torsion-free modules M and N over a complete domain are anti-isomorphic, then M and N are free finitely generated modules. The isomorphism problem can be studied for automorphism groups of modules (as well as for other algebraic structures). The following question is a fundamental and very difficult problem: when are modules isomorphic if their automorphism groups are isomorphic? Rickart [253] has obtained an affirmative answer for vector spaces over division rings of characteristic = 2; Leptin [195] and Liebert [205] have obtained an affirmative answer for Abelian p-groups (p = 2). Corner and Goldsmith [49] have proved that if M and N are reduced p-adic, torsion-free modules with Aut(M ) ∼ = Aut(N ) (p = 2), then M ∼ = N. Another direction of the study of the isomorphism problem for endomorphism rings is indicated by the papers of Hausen and Johnson [132], Hausen, Praeger, and Schultz [133], and Schultz [268]. These papers contain information on Abelian p-groups G and H that are isomorphic to the radicals J(End(G)) and J(End(H)) (as rings without unity). For example, whether G and H are isomorphic and whether there exists an isomorphism between G and H that induces a given isomorphism of radicals. If I is an arbitrary ideal of the endomorphism ring of some module M and I is contained in the radical J(End(M )), then 1 + I is a normal subgroup of the group Aut(M ) (Problem 6 (Part I) is connected to this subgroup). Schultz [269] used this property, and the results of [133, 268] extend the Leptin–Liebert theorem [195, 205] to the case p = 2. Problem 19. Find classes of modules M such that the isomorphism J(End(M )) ∼ = J(End(N )) implies the isomorphism M ∼ = N (the existence of an isomorphism M ∼ = N that induces the original isomorphism of radicals). Problem 20. For which modules M and N does the isomorphism Aut(M ) ∼ = Aut(N ) imply the isomorphism M ∼ = N? In these two problems, we call special attention to Warfield modules and torsion-free modules over complete domains. The following problem is of a more general nature. 3345
Problem 21. For various modules M and N , study all isomorphisms between End(M ) and End(N ), J(End(M )) and J(End(N )), Aut(M ) and Aut(N ), respectively. Problem 22. Characterize modules M with the following property. For every module N , the existence of a (resp., topological) isomorphism from End(M ) onto End(N ) implies the existence of an isomorphism from M onto N (resp., implies the existence of an isomorphism from M onto N which induces the original isomorphism of endomorphism rings).
Chapter 8 MODULES WITH MANY ENDOMORPHISMS OR AUTOMORPHISMS In Chapter 8, we consider the following topics: transitive and fully transitive modules (Sec. 37); transitivity over torsion and transitivity mod torsion (Sec. 38); the equivalence of transitivity and full transitivity (Sec. 39). Let x and y be two nonzero elements of some module M . In contrast to the case of vector spaces, it is not necessarily possible to map x onto y by some endomorphism or an automorphism of the module M . One of the obvious obstructions is that either the height sequences U (x) and U (y) can be incomparable or U (x) > U (y). Even if we remove this obstruction, the required endomorphism or automorphism does not necessarily exist. In the chapter, we consider modules that have sufficiently many (in a certain sense) endomorphisms or automorphisms. Considerable attention is given to mixed completely decomposable modules. 37.
Transitive and Fully Transitive Modules
We define transitive and fully transitive modules and present some relatively elementary results on such modules. We recall some notation. Let M be a module, and let x ∈ M . Then o(x) and h(x) are the order and height of the element x in M , respectively. We assume that the height means the “generalized height,” i.e., the height in the sense of Sec. 28. The height sequence U (x) of the element x in M is the sequence consisting of ordinal numbers and the symbol ∞ (h(x), h(px), h(p2 x), . . .). The following partial order can be defined on the set of height sequences: U (x) ≤ U (y) ⇐⇒ h(pi x) ≤ h(pi y) for i = 0, 1, 2, . . .. By definition, the greatest lower bound inf(U (x), U (y)) is the sequence (σ0 , σ1 , σ2 , . . .), where σi = min(h(pi x), h(pi y)). Let D be the maximal divisible submodule of the module M , and let D = 0. We assumed earlier that h(x) = ∞ for every x ∈ D and σ < ∞ for every ordinal number σ. We introduce a new symbol ∞+ , which is needed for a correct formulation of the main definitions and the reduction of the study to the case of reduced modules. We set h(0) = ∞+ and ∞ < ∞+ . The symbol ∞+ is used only for distinguishing the height of the zero element of the module M and the height of a nonzero element in D; if M is a reduced module, then it is not necessary to use the symbol ∞+ . Definition 37.1. Let M be a module. We say that (1) M is a fully transitive module if for all x, y ∈ M with U (x) ≤ U (y), there exists an endomorphism of the module M that maps x onto y; (2) M is a transitive module if for all x, y ∈ M with U (x) = U (y), there exists an automorphism of the module M that maps x onto y. The restrictions to the height sequences in (1) and (2) are necessary in the sense that for every endomorphism (automorphism) α of the module M , we have U (x) ≤ U (αx) (U (x) = U (αx)), x ∈ M . 3346
In the process of verifying the transitivity or full transitivity, we can assume (if is convenient) that x and y are nonzero elements. We often deal with the following situation (or a situation close to the it). Let M = A ⊕ N and let x, y ∈ A. If it proved that there exists an endomorphism (resp., an automorphism) α of the module A with αx = y, then we assume that α is an endomorphism (resp., an automorphism) of the module M ; in addition, we assume that α coincides on N with the identity mapping. We also note that the inequality U (x) ≤ U (y) always implies the inequality o(x) ≥ o(y). Proposition 37.2. Any divisible module is transitive and fully transitive. Proof. First, we assume that D is a divisible torsion-free module. The following stronger assertion holds. For any two nonzero elements x and y in D, there exists an automorphism of the module D that maps x onto y. If the rank of D is 1, then D = K. The endomorphisms of the module K coincide with multiplications by elements of K (Example 12.3 (Part I)). Consequently, all nonzero endomorphisms of the module K are automorphisms. Now it remains to use the relation y = (yx−1 )x. An arbitrary divisible torsion-free module D is a K-space. Consequently, Kx ⊕ G = D = Ky ⊕ H, where the modules G and H are isomorphic. It was just mentioned that there exists an isomorphism β : Kx → Ky with βx = y. Let γ be some isomorphism of G onto H. Then (β, γ) is an automorphism of the module D that maps x into y. Let D be a divisible primary module. We have that the height sequence U (x) of the element x ∈ D of order pk has the form (∞, . . . , ∞, ∞+ , . . .), where the symbol ∞+ starts from the position k + 1. We take any two elements x, y ∈ D with U (x) ≤ U (y) (resp., U (x) = U (y)). Let D = R(p∞ ) and let c1 , c2 , . . . , cn , . . . be the generator system for D considered in Sec. 4 (Part I). Since U (x) ≤ U (y), we have s ≤ t, where ps = o(y) and pt = o(x). Consequently, x, y ∈ Rct , x = uct , and y = pk vct , where k ≥ 0 and the elements u and v of R are invertible in the ring R. The multiplication of the module R(p∞ ) by the element pk vu−1 maps x into y. If U (x) = U (y), then k = 0, and we have an automorphism of the module R(p∞ ). If D is an arbitrary divisible primary module, then it follows from the proof of Theorem 6.3 (Part I) that there exist decompositions P1 ⊕ G = D = P2 ⊕ H, where x ∈ P1 , y ∈ P2 , P1 , P2 ∼ = H. By what was said above, there exists a homomorphism = R(p∞ ), and G ∼ P1 → P2 that maps x onto y; this yields an endomorphism of the module D mapping x into y. If height sequences coincide, we obtain the required automorphism. Now we assume that D is a divisible mixed module and D = Dt ⊕ D0 , where Dt is a primary module and D0 is a torsion-free module. Let x, y ∈ D, and let U (x) ≤ U (y) (U (x) = U (y)). We have x = x1 + x2 and y = y1 + y2 , where x1 , y1 ∈ Dt and x2 , y2 ∈ D0 . We consider two possible cases. Case (a): x2 = 0. If y2 = 0, then U (y2 ) = (∞, ∞, . . .) and U (x1 ) ≤ U (y) = U (y2 ), which is impossible. Consequently, y2 = 0 and x, y ∈ Dt . There exists an endomorphism (an automorphism) α of the module Dt and D with αx = y. Case (b): x2 = 0. There exists a submodule G such that D = Dt ⊕ G and x ∈ G (see the remark after of Theorem 6.1 (Part I)). If y2 = 0, then we similarly have D = Dt ⊕ H and y ∈ H. Then our argument is similar to the argument used in (a). The relation y2 = 0 is possible only if U (x) < U (y). Since Rx ∼ = R, there exists a homomorphism ϕ : Rx → D with ϕx = y. It extends to an endomorphism of the module D, since D is injective. Let A be a direct summand of the module M , and let x ∈ A. The height sequences of the element x with respect to A and M are equal. Consequently, we usually write U (x) instead of UA (x) or UM (x). Lemma 37.3. A direct summand A of a fully transitive module M is a fully transitive module. Proof. We denote by π the projection M → A. Let x, y ∈ A and let U (x) ≤ U (y). Consequently, αx = y for some α ∈ End(M ). Then (πα|A )x = y and πα|A ∈ End(A). Proposition 37.4. Let M be a module and let D be the maximal divisible submodule in M . The module M is transitive (resp., fully transitive) iff the factor module M/D is transitive (resp., fully transitive). 3347
Proof. We assume that M is a transitive module. We have M = A ⊕ D for some A. Since M/D ∼ = A, we need to prove that A is transitive. If x, y ∈ A and U (x) =U (y),then we take an automorphism α of the β ϕ with respect to a given decomposition module M with αx = y. We represent α as the matrix 0 γ of the module M (as in Proposition 2.4 (Part I)), where β (resp., γ) is an endomorphism of the module A (resp., D) and ϕ is a homomorphism from A into D. It is easy to verify that β is an automorphism of the module A and βx = (απ)x = π(αx) = πy = y, where π is the projection from M on A. Conversely, let A be a transitive module, x, y ∈ M , and U (x) = U (y). We have x = x1 + x2 and y = y1 + y2 , where x1 , y1 ∈ A and x2 , y2 ∈ D. We assume that x1 = 0. Then y1 = 0. Otherwise, U (x2 ) = U (y) = U (y1 ), which is a contradiction. Thus, x, y ∈ D. By Proposition 37.2, there exists an automorphism of the module M that maps x onto y. If x1 = 0, then we also have y1 = 0. We have the decompositions B ⊕ D = M = C ⊕ D, where x ∈ B and y ∈ C. It follows from B ∼ =A∼ = C that there exists an isomorphism α : B → C with αx = y. Assuming that α coincides on D with the identity mapping, we obtain the required automorphism. We pass to full transitivity. It follows from Lemma 37.3 that it remains to prove that full transitivity of the module A implies full transitivity of the module M . The proof is similar to the proof in the case of transitive modules. Up to the end of the chapter, we assume that M is a reduced module. We return to our agreement that h(0) = ∞. We present examples of transitive and fully transitive modules in each of the three main classes of modules: primary modules, torsion-free modules, and mixed modules. First, we consider cyclic primary modules. If x ∈ R(pm ) and o(x) = pk , then k is denoted by e(x). We have the obvious relation e(x) + h(x) = m. Now we assume that A = Ra and B = Rb, where the first module is isomorphic to R(pm ) and the second module is isomorphic to R(pn ). Let x ∈ A, y ∈ B, and U (x) ≤ U (y). We verify the existence of a homomorphism ϕ : A → B with ϕx = y. We have h(x) ≤ h(y) and e(y) ≤ e(x). We have x = ps ua and y = pt vb, where s ≤ t and u and v are invertible elements in R. The relations e(x) + s = m and e(y) + t = n hold. Furthermore, we have e(pt−s u−1 vb) = n − t + s = e(y) + t − t + s = e(y) + s ≤ e(x) + s = m. Thus, we obtain
o(pt−s u−1 vb) ≤ o(a). We have obtained that the mapping ra → rpt−s u−1 vb (r ∈ R) is a homomorphism A → B that maps x into y. For U (x) = U (y), we have h(x) = h(y) and e(x) = e(y). Therefore, o(a) = o(b) and A ∼ = B. The same mapping is already an isomorphism. Proposition 37.5. A primary module M without elements of infinite height is transitive and fully transitive. Proof. Let x, y ∈ M and let U (x) = U (y). It follows from Corollary 7.6 (Part I) that we can assume that M is a direct sum of finitely many cyclic modules. If x and y are of order p, then Ra ⊕ G = M = Rb ⊕ H, where x ∈ Ra and y ∈ Rb (Corollary 7.5 (Part I)). It follows from the argument presented before the proposition that there exists an automorphism of the module M that maps x into y. Let o(x) = pk , k ≥ 2. Since U (px) = U (py), by induction it follows that there exists an automorphism α with α(px) = py. If αx = y , then U (y ) = U (y). It suffices to prove the existence of an automorphism that maps y into y. We have y = y + z, where pz = 0 and h(y) ≤ h(z). There exists a decomposition M = Ra1 ⊕ · · · ⊕ Ras with z ∈ Ras . We have y = y1 + · · · + ys with yi ∈ Rai . We need to prove the existence of an automorphism that maps y onto y + z. We assume that ys = 0. We choose a subscript i with h(yi ) ≤ h(z). For simplicity, assume that i = 1. Let y1 = pm ua1 , and let z = pn vas , where n ≥ m and u and v are invertible elements of the ring R. We have the decomposition Ra1 ⊕ Ras = R(a1 + pn−m u−1 vas ) ⊕ Ras . 3348
There exists a mapping α such that α maps a1 into a1 + pn−m u−1 vas and α coincides with the identity mapping on the summands Ra2 , . . . , Ras . It is clear that α is an automorphism of the module M and αx = y. Let ys = 0. If ys +z = 0, then, as above, we can construct an automorphism β with β(y +z) = y . Then β −1 is the required automorphism. If ys + z = 0, then h(ys ) = h(ys + z). We denote by α the mapping that coincides with the identity mapping on Ra1 , . . . , Ras−1 and maps ys into ys + z. Let us prove the full transitivity. Let x, y ∈ M , U (x) ≤ U (y), and M = Ra1 ⊕ · · · ⊕ Ras . It suffices to consider the case where y is contained in one of the summands Rai . For example, let y ∈ Ra1 . We have x = x1 + x , where x1 ∈ Ra1 and x ∈ Ra2 ⊕· · ·⊕Ras . If h(x1 ) ≤ h(y), then there exists an endomorphism α of the module M with the properties αx1 = y and αx = 0. Otherwise, U (x) = U (x ), U (x ) ≤ U (y), and U (y + x ) = U (x ). Let γ be an automorphism of the module M with γx = y + x . The composition of γ and the projection M → Ra1 yields the required endomorphism. Hill [138] has proved that a totally projective module is transitive and fully transitive. Griffith [121] found other classes of transitive fully transitive primary modules. In works of Megibben [223] and Carroll and Goldsmith [36], primary modules without any transitivity property are constructed. Kaplansky [160] has proved that if 2 is an invertible element in R, then transitivity always implies full transitivity in the primary case. Corner [46] has proved the independence of these two transitivity notions for Abelian 2-groups. The papers of Megibben [225] and Goldsmith [113] also contain results on independence. For a nonzero element of some torsion-free module, the relation h(px) = h(x) + 1 holds. The height h(x) contains complete information about the height sequence U (x). For example, U (x) ≤ U (y) ⇐⇒ h(x) ≤ h(y). Proposition 37.6. Every torsion-free module M over a complete discrete valuation domain R is transitive and fully transitive. Proof. In fact, the full transitivity was proved in the beginning of Sec. 17 (Part I). Let x, y ∈ M and let h(x) = h(y). If the elements x and y are linearly dependent, then M = A ⊕ G, where x and y are contained in the module A isomorphic to R. There exists an automorphism of the module A that maps x into y (this follows from the arguments from Sec. 17 (Part I)). We assume that x and y are linearly independent elements. Then the pure submodule generated by x and y is equal to A ⊕ B, where A and B are isomorphic to R. By Corollary 11.5 (Part I), M = A ⊕ B ⊕ H. We have A1 ⊕ B1 = A ⊕ B = A2 ⊕ B2 , where x ∈ A1 and y ∈ A2 (see the beginning of Sec. 17 (Part I)). Now it is easy to construct an automorphism that maps x into y. A transitive torsion-free module M over an arbitrary (not necessarily complete) domain is fully transitive. Indeed, if x, y ∈ M and h(x) ≤ h(y), then y = pk z, where h(z) = h(x). Let α be an automorphism that maps x into z. Then (pk α)x = y. Exercises 6–8 contain more detailed results on the transitivity for torsion-free modules. We pass to mixed modules. There is an obvious necessary condition of transitivity (full transitivity) for mixed modules. Precisely, the torsion submodule t(M ) of a transitive (resp., fully transitive) of the mixed module M must be transitive (resp., fully transitive). This condition is certainly not sufficient. This can be easily verified by using the functor G from Theorem 22.2 (Part I). Let X be a torsion-free module that is not fully transitive (and there are many such modules). Since G(X)1 ∼ = X, the mixed module G(X) is neither fully transitive nor transitive. In addition, t(G(M )) is a direct sum of cyclic modules. Let M be a mixed module such that t(M ) has no elements of infinite height. It follows from Corollary 7.6 (Part I) that for all elements a1 , . . . , an ∈ t(M ), there exists a decomposition t(M ) = A ⊕ N , where A is a direct sum of finitely many cyclic modules and a1 , . . . , an ∈ A. By Theorem 7.2 (Part I), the module A is a direct summand of the module M . We will often use this property. Theorem 37.7 (Files [74]). Let M be a mixed module of rank 1. We assume that the torsion submodule t(M ) has no elements of infinite height. Then M is a transitive module and a fully transitive module. 3349
Proof. If M/t(M ) ∼ = R, then M = A ⊕ t(M ), where A ∼ = R. We leave this verification to the reader (see Exercise 1). Let M/t(M ) = K. First, we prove the full transitivity. Let x, y ∈ M and let U (x) ≤ U (y). We have sy = rx + c, where s, r ∈ R and c ∈ t(M ). Using the inequality h(x) ≤ h(y), we can restrict ourselves to the case s = 1. Thus, we have y = rx + c. We note that U (x) ≤ U (c). The module multiplication by r maps x into rx. Therefore, the proof reduces to the case where y ∈ t(M ). If x ∈ t(M ), then, as was mentioned before the theorem, x and y are contained in a direct summand of the module M that is a direct sum of cyclic primary modules. In this case, we can use Proposition 37.5. Therefore, assume that x is not contained in t(M ) and y is contained in t(M ). We have M = N ⊕ A, where y ∈ A and A is the direct sum of finitely many cyclic primary modules; in particular, pn A = 0 for some n. According to the decomposition, we have x = x0 + a. Since the module M/t(M ) is divisible, x0 = pn x1 + b, x1 ∈ N , b ∈ t(N ). We clain that U (b + a) ≤ U (y). Since h(pi (b + a)) = min(h(pi b), h(pi a)), it suffices to verify that h(pi b) ≤ h(pi y) if h(pi a) > h(pi y) and pi y = 0. In this case, min(h(pi+n x1 + pi b), h(pi a)) = h(pi x) ≤ h(pi y) < h(pi a). It follows from h(pi y) < n that h(pi+n x1 + pi b) = h(pi b)
and h(pi b) = h(pi x) ≤ h(pi y),
which is what was required. Now we have N = N0 ⊕ A0 , where A0 is a direct sum of finitely many cyclic modules and b ∈ A0 . An element b + a can be mapped into y by some homomorphism ϕ from A0 ⊕ A into A. Assuming that ϕ annihilates N0 , we extend ϕ to an endomorphism of the module M . Then ϕx = pn ϕ(x1 ) + ϕ(b + a) = y, since pn A = 0. Consequently, M is a fully transitive module. We verify that M is transitive. We assume that U (x) = U (y), x, y ∈ M . Then either x, y ∈ t(M ) or the orders of the elements x and y are infinite. If x, y ∈ t(M ), then we can use Proposition 37.5 once again. Let x, y ∈ / t(M ). Then y = rx + c, where r ∈ R and c ∈ t(M ). In addition, the element r is invertible, since U (x) = U (y). We use induction on the order of the element c. For c = 0, the module multiplication by r is an automorphism that maps x into y. We assume that c = 0. We have U (px) = U (py), py = rpx + pc, and o(pc) < o(c). Therefore, we inductively obtain an automorphism α with α(px) = py. We set x = αx. Now it suffices to find an automorphism of the module M that maps x onto y. Since px = py, we have y = x + a with pa = 0. Let M = N ⊕ A, where A is a direct sum of finitely many cyclic modules and a ∈ A. According to the decomposition, we have x = x1 + a1 and y = x1 + (a1 + a). If h(x ) = h(x1 ), then U (x1 ) ≤ U (a), and there exists a ϕ ∈ Hom(N, A) with ϕ(x1 ) = a, since M is a 1 ϕ of the module N ⊕ A that maps x to y. fully transitive module. In this case, the automorphism 1 1 To complete the proof, we can assume that h(x ) = h(a1 ) < h(x1 ). Since h(a1 ) = h(x ) = h(y) = min(h(x1 ), h(a1 + a)) < h(x1 ), we have h(a1 + a) = h(a1 ). Consequently, U (a1 ) = U (a1 + a). We choose an automorphism γ of the module A that maps from a1 into a1 + a. Then the automorphism (1, γ) of the module M = N ⊕ A maps x into y, which is what was required. For a mixed module of rank 1 with totally projective torsion submodule, Files [74] has obtained a result similar to Theorem 37.7. Theorem 37.7 does not hold even for modules of rank 2. Files [74] has constructed a p-adic module M such that M/t(M ) is a divisible module of rank 2, t(M ) has no elements of infinite height, but M is neither transitive nor fully transitive. As for other torsion submodules t(M ) in Theorem 37.7 and the necessary condition indicated before the theorem, we make the following remark. 3350
The same paper [74] contains the construction of a module M of rank 1 such that t(M ) has one of the transitivity properties but M is neither transitive nor fully transitive. It seems that topics related to transitivity (full transitivity) are interesting for Warfield modules. Files [74] has proved that a Warfield module of rank ≤2 is transitive and fully transitive. For larger ranks, this is not true. Hill and Ullery [146] proved that every Warfield module satisfies some weak transitivity. They considered four different classes of Warfield modules of rank 3 that are not transitive. Exercise 1. If P is a primary module, then the module R ⊕ P is transitive (resp., fully transitive) iff the module P is transitive (resp., fully transitive). Exercise 2. Let M = A ⊕ P , where A and P are a fully transitive, torsion-free module and a primary module, respectively. Then M is a fully transitive module. Exercise 3. Study the situation with transitivity for the module A ⊕ P , where A is a transitive torsionfree module and P is a transitive primary module. Exercise 4. Let M = P ⊕ N , where P is a primary module without elements of infinite height and N is some transitive (resp., fully transitive) module. Is the module M transitive (resp., fully transitive)? Exercise 5 (Files [74]). Any Warfield module of rank 1 is transitive and fully transitive. Exercise 6. Let M be a fully transitive, torsion-free module and let T be the center of the ring End(M ). Then T is a discrete valuation domain and every T -submodule in M of finite or countable rank is free. Exercise 7. For a torsion-free module M of finite rank, the following conditions are equivalent: (1) M is a transitive module; (2) M is a fully transitive module; (3) the center T of the ring End(M ) is a discrete valuation domain, EndR (T ) = EndT (T ), and M is a free T -module. Exercise 8. Let M and N be two fully transitive torsion-free modules. Then every topological isomorphism End(M ) → End(N ) is induced by some isomorphism M → N . 38.
Transitivity over Torsion and Transitivity Mod Torsion
In the remaining sections, the studies are mainly devoted to mixed modules. In this section, we introduce two weak notions of transitivity, which are specially adapted for mixed modules. Some of the results presented are general and other results are related to the class of completely decomposable modules; we study these modules in sufficient detail. Definition 38.1. Let M be a module. We say that (1) M is transitive over torsion if for all x ∈ M and y ∈ t(M ) with U (x) = U (y), there exists an automorphism of the module M that maps x into y; (2) M is transitive mod torsion if for all x, y ∈ M with U (x) = U (y), there exists an automorphism α with αx − y ∈ t(M ). Modules that are fully transitive over torsion or mod torsion are defined similarly. In (1) and (2), it is necessary to write U (x) ≤ U (y) and “an endomorphism” instead of U (x) = U (y) and “an automorphism,” respectively. We note that M is transitive mod torsion (resp., fully transitive mod torsion) iff pk (αx) = pk y for some automorphism (resp., endomorphism) α of the module M and some k ≥ 0 in all cases where U (x) = U (y) (resp., U (x) ≤ U (y)). The proof of the following simple result is similar to the proof of Lemma 37.3. Lemma 38.2. If M is a fully transitive over torsion (resp., mod torsion) module, then every direct summand of the module M is fully transitive over torsion (resp., mod torsion). 3351
Proposition 38.3. A module M is fully transitive iff M is fully transitive over torsion and fully transitive mod torsion. Proof. Assume that the module M is fully transitive over torsion and mod torsion. If x, y ∈ M and U (x) ≤ U (y), then there exist an endomorphism α of the module M and the element c ∈ t(M ) such that αx = y + c (since M is fully transitive mod torsion). Consequently, U (x) ≤ U (αx − y) = U (c). Using the property of full transitivity over torsion, we can find an endomorphism β with βx = c. Then (α −β)x = y. The converse implication is trivial. The analogue of Proposition 38.3 for transitivity does not hold (see Exercise 2). We have only the following result. Lemma 38.4. A module M is transitive iff M is transitive mod torsion and the following condition holds: in each case where elements x ∈ M and y ∈ M [p] satisfy the relation h(x) = h(x + y), there exists an automorphism of the module M that maps x into x + y. Proof. We assume that M is transitive mod torsion and satisfies the indicated condition. Let z and y be two elements of M that have the same height sequence. Then y = αz + c for some automorphism α and the element c of finite order. We set x = αz. It suffices to prove that x can be mapped into x + c by some automorphism. For c = 0, we take the identity mapping as such an automorphism. Applying the induction on the order of the element c, we obtain β(px) = p(x + c), where β is some automorphism. Note that U (βx) = U (x) = U (x + c) = U (βx + (x + c − βx)) and p(x+c−βx) = 0. By assumption, there exists an automorphism γ that maps βx into βx+(x+c−βx) = x + c. Then (βγ)x = x + c, which is what was required. The converse assertion is obvious. We pass to the main result of the section. Theorem 38.5 (Files [75]). A completely decomposable module is fully transitive mod torsion iff the module is transitive mod torsion. Proof. First, we assume that a completely decomposable module M is not fully transitive mod torsion. This means that there exist elements x, y ∈ M such that U (x) ≤ U (y) and, for all m, it is impossible to map pm x into pm y by any automorphism of the module M . Clearly, we can assume that there is a the decomposition M = M1 ⊕ M2 , where M1 is of rank 1 and y ∈ M1 . It is obvious that the order of y is infinite. Let x = x1 + x2 , where x1 ∈ M1 and x2 ∈ M2 . We assume that the coordinates of sequences U (x) and U (x1 ) coincide at an infinite number of positions, i.e., h(pk x) = h(pk x1 ) for an infinite set of integers k. It follows from U (x) ≤ U (y) that h(pk x1 ) ≤ h(pk y) for an infinite set of integers k. It is easy to show that upm y = vpn x1 for some invertible elements u, v ∈ R and nonnegative integers m and n with m ≤ n. The composition of the projection from M on M1 with the multiplication of M1 by the element vu−1 pn−m maps pm x into pm y. This contradiction proves that we can replace x and y by pk x and pk y for a sufficiently large integer k in order to assume that h(pi x1 ) > h(pi x) for all i ≥ 0. However, then U (x) = U (x − x1 ). Therefore U (x) ≤ U (x + y) = inf(U (x1 + y), U (x − x1 )) ≤ U (x − x1 ) = U (x). We have that x and x + y have the same height sequences. There is no automorphism α that maps pm x onto pm (x + y); otherwise, (α − 1)(pm x) = pm y, which is impossible. Therefore, M is not transitive mod torsion. Now we assume that M is fully transitive mod torsion. We can assume that M = M1 ⊕ · · · ⊕ Mn , where all Mi is of rank 1. Take elements x and y of infinite order with equal height sequences. We have x = x1 + · · · + xn ,
y = y1 + · · · + yn ,
xi , yi ∈ M,
i = 1, . . . , n.
We need to obtain an automorphism of the module M that maps pm x into pm y for some m ≥ 0. It is clear that for any two automorphisms α and β and an integer k ≥ 0, the elements x and y can be replaced 3352
by α(pk x) and β(pk y), respectively. We repeat such a procedure until it will be obvious that x can be mapped into y by some automorphism of the module M . First, we assume that there exists a positive integer m such that h(pk x1 ) > h(pk x) for all k ≥ m. Then h(pk x) = h(pk (x − x1 )) for all such integers k. We can enlarge m (if necessary) in order to assume that ϕ(pm (x − x1 )) = pm x1 for some ϕ : M2 ⊕ · · · ⊕ Mn → M1 . According to the decomposition M = M1 ⊕ (M2 ⊕ · · · ⊕ Mn ), the matrix
1 0 −ϕ 1
represents the automorphism α of the module M that maps pm x into pm (x − x1 ) ∈ M2 ⊕ · · · ⊕ Mn . Consequently, we can replace x and y by α(pm x) and pm y, respectively; therefore, we can assume that for x1 = 0, the coordinates in U (x) and U (x1 ) coincide at an infinite number of positions. We repeat this procedure for each of the remaining components x2 , . . . , xn (it is important that this procedure does not damage the results of our previous work), and then we make the same actions with each of the components y1 , . . . , yn of the element y. As a result, we obtain the situation where, for every i, the coordinates in the sequences U (x) and U (xi ) (resp., U (y) and U (yi )) coincide at an infinite number of positions if xi = 0 (resp., yi = 0). We assert that we can preserve this situation and additionally ensure the property that xi = 0 ⇐⇒ yi = 0, 1 ≤ i ≤ n. For example, assume that y1 = 0, but x1 = 0. Then the order of y1 is infinite. We denote by πi the projection from M on Mi . Since U (x) = U (y) ≤ U (y1 ) and M is fully transitive mod torsion, we can assume that y1 = ϕx for some ϕ ∈ End(M ). We have the relation y1 = (ϕπ1 )x2 + · · · + (ϕπ1 )xn . Since the rank of M1 is equal to 1, the coordinates of sequences U (y1 ) and U (y) coincide at an infinite number of positions and U (y) = U (x) ≤ U ((ϕπ1 )xi ), 2 ≤ i ≤ n, we have that U (y1 ) coincides with at least one sequence U ((ϕπ1 )xj ), j = 1, starting from some position. The matrix ⎞ ⎛ 1 0 0 ⎝ϕπ1 1 0⎠ 0 0 1 corresponding to the decomposition M = M1 ⊕ Mj ⊕ Mi i =1,j
represents an automorphism β that maps x into ((ϕπ1 )xj , x2 , . . . , xn ). We note that the order of the element x1 = (ϕπ1 )xj is infinite and that the coordinates of sequences U (x1 ) and U (x) coincide at an infinite number of positions. We can replace x by βx; therefore, we can assume that x1 = 0 provided y1 = 0. We continue similar replacements with respect to all remaining nonzero components of the element y; then we continue similar replacements with respect to every nonzero component of the element x. Thus, we obtain the situation described above. Now it is easy to find an integer m and an automorphism γ with γ(pm x) = pm y. First, note that if xi = 0, then h(pk xi ) ≤ h(pk yi ) for an infinite set of integers k (since U (x) = U (y) ≤ U (yi ) and the coordinates in U (xi ) and U (x) coincide at an infinite number of positions). Similarly, h(pk yi ) ≤ h(pk xi ) for an infinite set of integers k. Since the rank Mi is equal to 1 and xi , yi are elements of infinite order, it follows from easy arguments that pmi yi = pmi ui xi for some integers mi ≥ 0 and invertible elements ui ∈ R. For every j, let γ coincide on Mj with the multiplication by uj if xj = 0, and let γ coincide on Mj with the identity mapping if xj (equivalently, yj ) is equal to zero. Let m be the maximal among the integers mj with xj = 0. Then γ(pm x) = pm y, and the proof is completed. 3353
The following variants of full transitivity are very useful (especially in the topics related to full transitivity of direct sums). Let M and N be two modules. An ordered pair (M, N ) is said to be fully transitive if for all x ∈ M and y ∈ N with U (x) ≤ U (y), there exists a homomorphism ϕ : M → N with ϕx = y. The pair (M, N ) is said to be fully transitive over torsion if the indicated homomorphism exists for all elements x ∈ M and y ∈ t(N ) with U (x) ≤ U (y). In the previous section, it was proved that every pair of cyclic primary modules is full transitive (see also Exercise 1). Proposition 38.6. (1) A pair (M, N ) is fully transitive over torsion iff it is fully transitive over N [p] (the meaning of this is explained in the proof ). (2) A direct sum M = Mi is fully transitive over torsion iff the pair (Mi , Mj ) is fully transitive over torsion for all i, j ∈ I.
i∈I
Proof. (1) We assume that the pair (M, N ) is fully transitive over N [p], i.e., for all x ∈ M and y ∈ N [p], we have ϕx = y for some homomorphism ϕ : M → N provided that U (x) ≤ U (y). Now let U (x) ≤ U (y), where x ∈ M and y ∈ t(N ). By the use of the induction on the order of the element y, we can assume that ϕ(px) = py for some ϕ : M → N . Then y = y − ϕx ∈ N [p] and U (x) ≤ U (y ). Consequently, ψx = y , where ψ : M → N . Then (ϕ + ψ)x = y. (2) Let x ∈ M , y ∈ M [p], and let U (x) ≤ U (y). It suffices to consider the case where M = M1 ⊕· · ·⊕Mn . We have x = x1 + · · · + xn and y = y1 + · · · + yn , where xi , yi ∈ Mi . We have h(x) = h(xk ) for some k. Consequently, U (xk ) ≤ U (y) ≤ U (yi ) (1 ≤ i ≤ n), since py = 0. By assumption, ϕi (xk ) = yi for some homomorphisms ϕi : Mk → Mi . Then (πk (ϕ1 + · · · + ϕn ))x = y, where πk is the projection of M on Mk . By 1), the module M is fully transitive over torsion. The proof of the converse assertion is similar to the proof of Lemma 37.3; see also Lemma 38.2. Corollary 38.7. If M is fully transitive over torsion, then the module M is fully transitive over torsion s
for every cardinal number s. Corollary 38.8 (Files [75]). A completely decomposable module is fully transitive over torsion if the torsion submodule of the module does not have elements of infinite height or it is a Warfield module. Proof. It follows from Proposition 38.6 that it suffices to prove that the pair (M, N ) is fully transitive over N [p] provided the rank of M is equal to 1 and either t(M ) and t(N ) has no elements of infinite height or M is a Warfield module. We assume that x ∈ M , y ∈ N [p], and U (x) ≤ U (y). We consider the first case. First, we assume that the module M is split; let M = Ra ⊕ t(M ). According to the decomposition, we have x = x1 + x2 . Using the projection from M onto Ra or t(M ), we can assume that x ∈ Ra or x ∈ t(M ). In the first case, it is easy to construct a homomorphism ϕ : Ra → Ry with ϕx = y. If x ∈ t(M ), then the result follows from Proposition 37.5. For the nonsplit module M , the factor module M/t(M ) is a divisible module (it is isomorphic to K). We denote by n the height of the element y. We have x = pn+1 a + c, where a ∈ M and c ∈ t(M ). We embed the element c in a finitely generated direct summand P of the module M . Then U (c) ≤ U (y), since c and x have the same height. Consequently, y = ϕc for some ϕ : P → Rz, where pn z = y. The composition of the projection from M onto P with ϕ maps x into y. As for the second possibility, a more general result is true. Precisely, the pair (M, N ) is fully transitive. By Corollary 32.8 (b), Rx is a nice submodule in M and M/Rx is a totally projective module. Consequently, the mappings Rx → N , which do not decrease heights, can be extended to homomorphisms from M into N (by the same corollary). The last corollary is not true for arbitrary completely decomposable modules. For a primary module P , the transitivity (resp., full transitivity) over torsion of the module R ⊕ P is equivalent to the ordinary transitivity (resp., full transitivity). For some modules P , this leads us to simple examples of completely decomposable modules that are transitive but not fully transitive, and conversely; for example, see Exercise 1 from Sec. 37. 3354
Exercise 1. Verify that (M, N ) and (A, P ) are fully transitive pairs, where M , N , and P are primary modules such that M and N have no elements of infinite height and A is a torsion-free module. Exercise 2 (Files [74, 75]). There exists a mixed nontransitive module of rank 1 that is transitive over torsion and transitive mod torsion. the following conditions are equivalent: Exercise 3. For a pure submodule M of the R-module R, (1) M is a transitive module; (2) M is a fully transitive module; such that the R-modules M and T are isomorphic to each (3) there exists a pure R-subalgebra T in R other and T is a discrete valuation domain. A module M is said to be quasi-pure injective if every homomorphism A → M , where A is a pure submodule of the module M , can be extended to an endomorphism of the module M . Exercise 4 (Dobrusin [53]). A reduced module M is quasi-pure injective iff either M is pure-injective or M∼ S, where A is a pure fully invariant submodule of an adjusted pure-injective module, S is a =A⊕ n
and S is a discrete valuation domain. pure R-subalgebra in R, Exercise 5. Is a quasi-pure injective module transitive or fully transitive? 39.
Equivalence of Transitivity and Full Transitivity
We return to the initial notions of transitivity and use our previous work for proving the results on interrelations between two transitivity properties for some large classes of mixed modules (Theorem 38.5 is also related to this topic). In particular, we pay attention to transitivity and full transitivity of direct sums of copies of a single module. Theorem 39.1 (Files [75]). Let M be a completely decomposable module. If the torsion submodule t(M ) has no elements of infinite height or M is a Warfield module, then the following conditions are equivalent: (1) M is a fully transitive module; (2) M is a transitive module; (3) M is fully transitive mod torsion; (4) M is transitive mod torsion. Proof. Naturally, we have (1) =⇒ (3) and (2) =⇒ (4). The equivalence of conditions (3) and (4) follows from Theorem 38.5. In addition, the implication (3) =⇒ (1) follows from Corollary 38.8 and Proposition 38.3. It remains to verify the implication (4) =⇒ (2). We use Lemma 38.4. Up to the end of the proof, we assume that x ∈ M , y ∈ M [p], and h(x) = h(x + y). We need to obtain an automorphism of the module M that maps x into x + y. Assume that t(M ) contains no elements of infinite height. An element y can be embedded in a finitely generated direct summand C of the module M ; let M = N ⊕ C. In this decomposition, we have x = z + c. If h(z) ≤ h(c), then U (z) ≤ U (y). Therefore, y = ϕz for some ϕ : N → C by Corollary 38.8. The automorphism 1 ϕ 0 1 maps x into x + y. Consequently, we can assume that h(z) > h(c) = h(x); therefore, U (c + y) = U (c). There exists an automorphism γ of the module C that maps c onto c + y (Proposition 37.5). Then the automorphism (1, γ) of the module M maps x onto x + y, which is what was required. We assume that M is a Warfield module. Since M is a completely decomposable module, there exists a decomposition M = A ⊕ M such that M is a completely decomposable module, A is a module of rank 1, x = x1 +x2 , y = y1 +y2 , h(x) = h(x1 ), x1 , y1 ∈ A, and x2 , y2 ∈ M . We note that U (x1 ) ≤ U (y2 ). It follows from the proof of Corollary 38.8 that the pair (A, M ) is fully transitive. Consequently, we have y2 = ϕ(x1 ) 3355
1 ϕ for some ϕ : A → We replace x by the image x1 + x2 + y2 of x under the automorphism . 0 1 Thus, we reduce the original problem to the case where y2 = 0. In this situation, if h(x1 ) = h(x1 + y1 ), then x1 + y1 = γ(x1 ) for some γ ∈ Aut(A) (A is transitive by Exercise 5 from Sec. 37). Then the automorphism (γ, 1) maps x into x + y1 . Consequently, we can assume that h(x1 + y1 ) > h(x1 ) = h(x). Then h(x) = h(x2 ), since h(x + y1 ) = h(x). Now we can choose a summand A of rank 1 of the module M such that with respect to the new decomposition M = A ⊕ M , the element y1 is contained in M and the component of the element x in A has a height sequence ≤U (y1 ). Furthermore, we can repeat the argument from the beginning of the paragraph in order to obtain an automorphism of the module A ⊕ M that maps x onto x + y1 . M .
We pass to direct sums of fully transitive modules. We already considered them in Proposition 38.6 and Corollary 38.7. Proposition 39.2 (Files [75]). Let M be a completely decomposable module. If M is fully transitive mod torsion, then the module M is fully transitive mod torsion for every cardinal number s. s
Proof. It suffices to consider the case where the rank of the module M is finite. Using Lemma 38.2, we see that it suffices to prove the full transitivity of the module M1 ⊕ M2 , where M1 and M2 are two copies of a completely decomposable module of finite rank that is fully transitive mod torsion. We denote by τ : M2 → M1 the identity mapping. We assume that elements x, z ∈ M1 and y ∈ M2 satisfy U (x + y) ≤ U (z). We choose some decomposition M1 = M2 = A1 ⊕ · · · ⊕ An , where the ranks of all summands are equal to 1. According to the decomposition, we have x = x1 +· · ·+xn and y = y1 +· · ·+yn . We replace the elements x, y, and z by the elements pm x, pm y, and pm z for a sufficiently large integer m. k Thus, we obtain a partition {1, . . . , n} = I ∪ J such that h(pk xi ) > h(p yi ) for all k ≥ 0 provided i ∈ I, and U (xi ) ≤ U (yi ) provided i ∈ J. Let π be the projection M2 → Ai ⊆ M2 . It follows from properties i∈I
of the given partition that U (x + (πτ )y) = inf(U (x), U (y)) = U (x + y). Since M1 is fully transitive mod torsion, there exist ϕ ∈ End(M1 ) and m ≥ 0 such that ϕ(pm (x+(πτ )y)) = pm z. The endomorphism ϕ 0 πτ ϕ 0 of the module M1 ⊕ M2 maps pm (x + y) into pm z. Now we pass to the general case where z = z1 + z2 ∈ M , zi ∈ Mi , i = 1, 2, and U (x + y) ≤ U (z). Since U (z) ≤ U (z1 ), U (z2 ), it follows from the proved properties that there exist α1 , α2 ∈ End(M ) and m ≥ 0 with αi (pm (x + y)) = pm zi , i = 1, 2. Then (α1 + α2 )(pm (x + y)) = pm z.
The following result follows from Assertions 38.3, 38.7, and 39.2. Corollary 39.3. If M is a completely decomposable, fully transitive module, then the module
s
M is
fully transitive for every cardinal number s. There exist interesting interrelations between properties of full transitivity of a module M and tran sitivity of the sum M , s ≥ 2. In particular, the module M ⊕ M , called the square of the module M , s M . Other interrelations between these two objects are presented in Exercise 7 of can be considered as s
Sec. 14 (Part I) and Problem 4 (b) (Part I). 3356
Lemma 39.4. Let M be an arbitrary module. If
s
M (s ≥ 2) is a transitive module, then M is a fully
transitive module. Proof. Let x, y ∈ M with U (x) ≤ U (y). The elements (x, 0, 0, . . .) and (y, x, 0, . . .) of the module
s
M
have the same height sequences. Since this module is transitive, there exists an automorphism α that maps (x, 0, 0, . . .) onto (y, x, 0, . . .). The composition of the embedding M in the first summand M of the module M , the automorphism α, and the projection on M yields the required endomorphism of the s
module M that maps x into y. We have that transitivity of the square of a module implies full transitivity of this module. For a completely decomposable module, the converse assertion holds. Theorem 39.5 (Files [75]). A completely decomposable module is fully transitive iff the square of the module is transitive. Proof. By Lemma 39.4, we need to only verify the transitivity of the module M1 ⊕M2 provided M1 and M2 are two copies of some fully transitive, completely decomposable module. Let τ : M1 → M2 be the identity mapping. We note that M1 ⊕ M2 is fully transitive by Corollary 39.3; therefore, M1 ⊕ M2 is transitive mod torsion by Theorem 38.5. We verify that the direct sum satisfies the condition of Lemma 38.4. We assume that x, y ∈ M1 ⊕ M2 , h(x) = h(x + y) and py = 0. According to the decomposition, we have x = x1 + x2 and y = y1 + y2 . We can assume that h(x) = h(x1 ). Consequently, y2 = ϕ(x1 ) for some ϕ : M1 → M2 , since the direct sum is fullytransitive and U (x1 ) ≤ U (y2 ). We replace x by the image 1 ϕ x1 + x2 + y2 of x under the automorphism . This allows us to assume that y = y1 ∈ M1 in the 0 1 1 0 original problem. If h(x2 ) ≤ h(y1 ), then y1 = ψ(x2 ), where ψ : M2 → M1 and the automorphism ψ 1 maps from x into x + y1 . Otherwise, we have h(x2 ) > h(x) = h(x1 ). We can verify that U (x1 + y1 ) ≤ U (τ y1 ), since h(x1 + y1 ) = h(x1 ) = h(x) ≤ h(y1 ). In addition, U (τ x1 + x2 ) ≤ U (y1 ), since h(τ x1 + x2 ) = h(x1 ) = h(x) ≤ h(y1 ). By assumption, there exist homomorphisms ϕ : M1 → M2 and ψ : M2 → M1 such that ϕ(x1 + y1 ) = τ (y1 ) and ψ(τ x1 + x2 ) = y1 . It can be verified that the composition 1 τ 1 ϕ 1 −τ 0 1 ψ 1 + ψϕ 0 1 of automorphisms of the module M1 ⊕M2 maps x onto x+y1 . This proves the condition from Lemma 38.4. Files [75] notes that Theorem 39.1 holds for every Warfield module. In this direction, Hennecke and Str¨ ungmann [134] have proved that if 2 is an invertible element in R and M is a transitive module, then M is fully transitive provided that M is a completely decomposable module or the rank of the maximal divisible submodule of the module M/t(M ) does not exceed 1. The first theorem, which is similar to Theorem 39.5, was first proved for primary modules, by Files and Goldsmith [77]. In [134], quite general results in this topic are contained. Let M be either a completely decomposable module, or a Warfield module, or a module such that the maximal divisible submodule of the module M/t(M ) has rank at most 1. Then the following conditions are equivalent: (1) M is a fully transitive module; 3357
(2) (3) (4) (5)
s s s s
M is a fully transitive module for some cardinal number s > 1; M is a fully transitive module for all cardinal numbers s > 1; M is a transitive module for some s > 1; M is a transitive module for all s > 1.
This theorem can be applied to primary modules and torsion-free modules. It is not known whether for every module M , the module M is fully transitive iff M ⊕ M is transitive. Also, it is not known whether the square M ⊕ M is fully transitive iff M ⊕ M is transitive. A more general problem is the search for transitivity conditions for direct sums and products of modules. In the corresponding studies, the notion of a fully transitive pair is intensively used. In the more general form of a “fully transitive system of modules,” the notion appeared in papers of Grinshpon and Misyakov (see [125, 128]); Files and Goldsmith [77] define this notion later and independently. These topics and other aspects of the theory of fully transitive Abelian groups and modules are presented in the papers of Grinshpon and Misyakov [123–126, 128, 229]. A review (with proofs) of many results of these papers is presented in [127]. Exercise 1 (Files [75]). Prove that Theorem 39.1 is true for all Warfield modules. Exercise 2. For a completely decomposable module M , the following conditions are equivalent: (1) M is transitive mod torsion; (2) M is transitive mod torsion for every (equivalently, for some) cardinal number s. s
Exercise 3. (a) Let M = A ⊕ B, and let ϕ : A → B and ψ : B → A be two homomorphisms. Verify that 1 ϕ the matrix is an invertible matrix. ψ 1 + ψϕ α 1 + αβ is invertible. (b) If M = A ⊕ A and α, β ∈ End(A), then the matrix 1 β Exercise 4 (see [77, 125, 128, 229], Proposition 38.6 (2)). For a set of primary modules {Ai }i∈I , the following conditions are equivalent: (1) Ai is a fully transitive module; i∈I (2) Ai is a fully transitive module; i∈I
(3) the pair (Ai , Aj ) is fully transitive for all i, j ∈ I. Exercise 5 ([75, 125, 127]). Let {Ai }i∈I (|I| ≥ 2) be a set of torsion-free modules. The following conditions are equivalent: (1) Ai is a transitive (resp., fully transitive) module; i∈I (2) Ai is a transitive (resp., fully transitive) module; i∈I
(3) the pair (Ai , Aj ) is a fully transitive pair for all i, j ∈ I. Exercise 6. Let M be a transitive torsion-free module. Every element of the product
s
M can be
embedded in a direct summand such that the summand is isomorphic to M and the complement summand is a transitive module. Remarks. A vector space over a division ring can be called “n-transitive” for every n in the wellknown sense. The notions of a transitive module and a fully transitive module were introduced by Kaplansky. The first study in Kaplansky’s work [160] and subsequent papers related to transitivity dealt with Abelian p-groups (the corresponding bibliography is presented in Sec. 37). Furthermore, transitive 3358
and fully transitive torsion-free Abelian groups have been systematically studies in papers of professors and postgraduates of the algebra department of Tomsk State University (Russia), see [38–41, 53, 124, 125, 127, 167, 168, 173, 174]. The main results of papers [38–40], and [41] are also presented in [42]. Mixed fully transitive groups are studied in the papers cited before the exercises in Sec. 39. Sometimes the study of transitivity properties allows us to better understand the structure of concrete modules. In any case, the study is related to a common topic: interrelations between endomorphisms and automorphisms of algebraic structures. The study of characteristic and fully invariant submodules is related to the same topic. Kaplansky [160] presented the description of fully invariant submodules of primary fully transitive modules. In [76] and [193], fully invariant submodules of Warfield modules and balanced projective modules are characterized. Fully invariant subgroups and their lattices are studied in [123–126] (see also [127]). The problem of additive generating of the endomorphism ring by the automorphism group is also related to the same topic; the problem was discussed in Remarks to Chapter 3 (Part I). Problem 23. For which classes of modules are conditions (1)–(3) of Exercises 4 and 5 in Sec. 39 equivalent? Problem 24. For which classes of modules does the theorem of Hennecke and Str¨ ungmann hold (the theorem is formulated after the proof of Theorem 39.5)? In particular, determine interrelations between the following properties of an arbitrary module M : (1) M is a fully transitive module; (2) M ⊕ M is a fully transitive module; (3) M ⊕ M is a transitive module. Problem 25. For which modules is the transitivity property equivalent to the full transitivity property? Problem 26. Describe transitive modules of rank 1 and Warfield modules. Problem 27. Study characteristic submodules and the lattices of characteristic submodules in various classes of modules. A module M over a ring S is called a multiplication module if every submodule of M is equal to IM for some ideal I of the ring S. Problem 28. Study modules that are multiplication modules over their endomorphism rings or modules that are multiplication modules over the centers of their endomorphism rings.
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