indecomposable flat cotorsion modules - CiteSeerX

J. London Math. Soc. (2) 76 (2007) 797–811

e 2007 London Mathematical Society

C

doi:10.1112/jlms/jdm076

INDECOMPOSABLE FLAT COTORSION MODULES PEDRO A. GUIL ASENSIO and IVO HERZOG Abstract An additive functor from the category of flat right R-modules to the category of abelian groups is continuous if it is isomorphic to a functor of the form − R M , where M is a left R-module. It is shown that for any simple subfunctor A of − ⊗ M there is a unique indecomposable flat cotorsion module UR for which A(U ) = 0. It is also proved that every subfunctor of a continuous functor contains a simple subfunctor. This implies that every flat right R-module may be purely embedded into a product of indecomposable flat cotorsion modules. If CE(R) is the cotorsion envelope of RR and S = EndR CE(R), then a local ring monomorphism is constructed from R/J(R) to S/J(S). This local morphism of rings is used to associate a semiperfect ring to any semilocal ring. It also proved that if R is a semilocal ring and M a simple left R-module, then the functor − R M on the category of flat right R-modules is uniform, and therefore contains a unique simple subfunctor.

Introduction Let R be an associative ring with identity. The category Flat-R of flat right R-modules is an accessible category in the sense of [1, 15] and so admits a theory of purity (see, for example, [1, Chapter 2]). This theory of purity for accessible categories has also been outlined in the additive case in [6] under the terminology of locally finitely presented categories. In this theory, pure-injective objects of Flat-R correspond to flat right R-modules CR such that Ext1R (F, C) = 0 for every flat right R-module F (see [14]). Modules satisfying this homological condition are called cotorsion and were first introduced by Harrison [13] in the homological study of abelian groups. On the other hand, the result of Bican, El Bashir and Enochs [4] that every R-module admits a flat cover implies that every R-module admits a cotorsion envelope. Therefore, the category Flat-R has enough pure-injective objects and every object in it admits a pure-injective envelope. In this article, we continue the line of research developed in the papers [10, 12, 11], which attempts on the one hand to extend the classical theory of pure-injective R-modules to the class of flat cotorsion modules, and on the other hand, to apply these results to study the structure of the category Flat-R. Ziegler [19, Corollary 6.9] showed that there exist enough pure-injective indecomposable objects in the category Mod-R in the sense that every module is elementarily equivalent to a direct product of pure-injective indecomposable modules. Actually, the category Mod-R can be fully embedded in its (covariant) functor category Func(R-mod, Ab), which is a locally finitely presented Grothendieck category, where R-mod is the category of finitely presented left R-modules. Under this embedding, a module M is identified with the fp-injective functor M ⊗R −, and thus any module M purely embeds in a direct product of indecomposable pureinjective R-modules (see also the proof of [16, Proposition 9.24]). In this article, it is shown that every flat module may be purely embedded into a direct product of indecomposable flat cotorsion modules. However, we stress that the above embedding can no longer be used in our new setting, and therefore we need to develop new techniques which have their origin

Received 10 October 2005; published online 19 November 2007. 2000 Mathematics Subject Classification 03C60, 03C98, 16D40, 15D50, 16D90, 16E30, 16L30, 16P70, 18A30. The first author is partially supported by the DGI (BFM 2000-0346, Spain) and by the Fundaci´ on S´ eneca (PI-76/00515/FS/01). The second author is partially supported by NSF grants DMS-02-00698 and DMS-05-01207.

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PEDRO A. GUIL ASENSIO AND IVO HERZOG

in the work of Auslander [3] on morphisms determined by objects. An additive functor from the category of flat right R-modules to the category of abelian groups is continuous, that is, commutes with direct limits, if and only if it is equivalent to a functor of the form − R M , where M is a left R-module. We show that, given a simple subfunctor A ⊆ − ⊗ M of a continuous functor, there is (Theorem 3) a unique indecomposable flat cotorsion R-module UR such that A(U ) = 0. Next, we prove the following theorem, which allows us to show that any flat module admits a pure monomorphism into a direct product of indecomposable flat cotorsion modules. Theorem (Theorem 7, with C = Flat-R). Let M be a left R-module. If B : Flat-R → Ab is a subfunctor of − ⊗ M , then B contains a simple subfunctor. Let us denote by CE(RR ) the cotorsion envelope of RR , and let S be the endomorphism ring of CE(RR ). For any element r ∈ CE(RR ), there is a morphism fr : RR → CE(RR ) given by fr (1) = r. This morphism extends to a (non-unique) endomorphism fr ∈ S. We show in Theorem 17 that the rule r → fr induces a local monomorphism of rings ϕ : R/J(R) → S/J(S), where J(R) and J(S) denote, respectively, the Jacobson radical of R and that of S. We recall (see [5]) that a homomorphism of rings is called local if every preimage of a unit is also a unit. In particular, R is a semilocal ring if and only if S is a semilocal (see [5, Theorem 1]). By [10, Theorem 8], this is equivalent to saying that S is semiperfect. Therefore, we are outlining a very natural way to associate to any ring R, another ring S such that R is semilocal if and only if S is semiperfect, thus shedding new light on the study of semilocal rings (see, for instance, [9, Chapter 4]). Namely, finitely generated projective S-modules are the projective covers of finitely generated semisimple S-modules, and therefore they are direct sums of projective covers of simple modules. However, this is no longer true for the original ring R. This is the reason for the pathological behavior of finitely generated projective modules over a semilocal ring. We show in our last section that this is due to the fact that simple R-modules do not properly determine the class of indecomposable projective R-modules but, rather, that of indecomposable direct summands of the cotorsion envelope of R. Actually, we show that these indecomposable direct summands are just the flat covers of those simple modules. Moreover, we also prove that if M is a simple left module over a semilocal ring R, then the continuous functor − R M is uniform and thus contains a unique simple subfunctor. Throughout the paper, R will denote an associative ring with identity, and J the Jacobson radical of R. We denote by ℵ = |R| + ℵ0 the smallest infinite cardinal such that ℵ |R|. If FR is a flat R-module, then the cotorsion envelope of F is unique up to isomorphism over F and will be denoted by CE(F ). We recall that a homomorphism of modules f : M → N is called strongly pure if any homomorphism g from M to a cotorsion module extends to N (see [11]). We refer to [18, Section 3.4] for the basic properties of the cotorsion envelope of a module and to [11, Proposition 2] for the basic properties of strongly pure monomorphisms. 1. Simple functors Throughout this section, C will denote a full additive subcategory of Flat-R closed under direct limits and cotorsion envelopes. An additive functor X : C → Ab is called continuous if it respects direct limits, X(lim Fi ) = lim X(Fi ). →

→

If X : Flat-R → Ab is a continuous functor on Flat-R, then X is naturally isomorphic to a functor of the form − R M . Indeed, as every flat module is a direct limit of free modules

INDECOMPOSABLE FLAT COTORSION MODULES

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of finite rank, a continuous functor is completely determined by the value X(RR ) = R M , which has a natural left R-module structure, induced by the endomorphism ring of the object RR ∈ Flat-R. In this way, the category of continuous functors on Flat-R is equivalent to the category R-Mod of left R-modules. It is evident that if M is a left R-module, then the restriction (− ⊗ M )|C to C ⊆ Flat-R of the continuous functor − R M on Flat-R yields a continuous functor on C. Let A, B : C → Ab be two functors. Then A ⊆ B will denote that A is a subfunctor of B. We shall invoke the following observation without comment. Lemma 1. Let I be a directed partial order and X : I → C a directed system, with structural morphisms {fij }ij . Suppose that B : C → Ab is a subfunctor of a continuous functor on C. If a ∈ B(X(i)) is such that for every fij , B(fij )(a) = 0, then B(fi )(a) = 0, where fi : X(i) → lim X(i) is the induced morphism to the direct limit. →

Proof. If B is itself continuous, then the conclusion follows from the universal property of the direct limit. Since B is a subfunctor of a continuous functor B , it inherits this property from B . Let B : C → Ab be an additive functor. If FR ∈ C and a ∈ B(F ), then the subfunctor of B generated by a on C is given by Ba (G) = B(Hom(F, G))a = {B(f )a | f : F → G}, for G ∈ C. An additive functor B : C → Ab is called finitely generated on C if there is a module FR ∈ C and a ∈ B(F ) is such that B = Ba . Remark. Let C be a small additive category and F : C → Ab a (covariant) additive functor. It is easy to check that F is a finitely generated functor according to this definition if and only if it is a finitely generated object in the Grothendieck functor category Func(C, Ab). Note, however, that Flat-R is not a small category and, therefore, Func(Flat-R, Ab) may no longer be a category. Lemma 2. Let M be a left R-module. If B : C → Ab is a non-zero subfunctor of (− ⊗ M )|C , then B contains a non-zero subfunctor Ba ⊆ B finitely generated on C by a ∈ B(C), where C is a cotorsion module in C. Proof. Let F ∈ C be an object for which there is a nonzero b ∈ B(F ). The cotorsion envelope of F , f : F → C, belongs to C by definition. Since the cotorsion envelope is a pure monomorphism, the morphism f ⊗ 1M : F ⊗ M → C ⊗ M of abelian groups is a monomorphism. Because B is a subfunctor of − ⊗ M on C, B(f ) : B(F ) → B(C) is also a monomorphism. Thus a = B(f )(b) is a non-zero element of B(C). The subfunctor Ba ⊆ B satisfies the conclusion of the lemma. Recall that a non-zero functor B : C → Ab is simple if it has no non-trivial subfunctors. Clearly, such a functor is generated by any non-zero element a ∈ B(F ), where F ∈ C. Theorem 3. Let M be a left R-module. If A : C → Ab is a simple subfunctor of (− ⊗ M )|C , then there is an indecomposable cotorsion module UR ∈ C such that A(U ) = 0. Furthermore, if FR ∈ C, then a morphism f : U → F is a split monomorphism if and only if A(f ) : A(U ) → A(F ) is non-zero. Such a morphism exists whenever A(F ) = 0.

800


Proof. By Lemma 2, A is generated on C by an element a ∈ A(C) ⊆ C ⊗ M , with CR ∈ C a cotorsion module. Since (− ⊗ M )|C is continuous, Zorn’s lemma may be used to provide a pure submodule G ⊆ C, maximal with respect to the condition that C/G ∈ C and that b = (pG ⊗ 1M )(a) is a non-zero element of (C/G) ⊗ M , where pG : C → C/G is the natural quotient map. Clearly, b ∈ A(C/G). As A is simple, it is also generated on C by b ∈ (C/G) ⊗ M . Thus a = A(f )(b) for some f : C/G → C. The composition g = f ◦ pG : C → C is an element of the endomorphism ring S = EndR C that contains G in its kernel, and a = A(g)(a). By [10, Proposition 3], there is an idempotent k : C → C such that k ∈ Sg and 1 − k ∈ S(1 − g). The latter relation implies that A(k)(a) = a; the former that the kernel of k contains G. By maximality of G, we see that G = Ker k is a direct summand and that U = C/G is indecomposable. Now suppose that FR ∈ C and that there is a morphism f : U → F such that A(f ) : A(U ) → A(F ) is non-zero. Let a ∈ A(U ) be such that b = A(f )(a) ∈ A(F ) is non-zero. Because A is simple, the element b ∈ A(F ) generates A on C, so there is a morphism g : F → U such that A(g)(b) = a. Thus A(g ◦ f )(a) = a, which implies that 1 − gf is not a unit in the ring T = EndR U . By [10, Corollary 7], the ring T is local, so that g ◦ f ∈ T is a unit, and the morphism f : U → F has a retraction. Let M be a left R-module with endomorphism ring S = EndR M (acting on the right). Denote by IS the minimal injective cogenerator of the category Mod-S of right S-modules. The local dual of M is the right R-module M := HomS (R MS , IS ), with action defined by (f r)(m) := f (rm). The local dual M is a pure-injective module [3, Proposition 10.1c], so it is cotorsion. Because the kernel of the flat cover g : FC(M ) → M is cotorsion, the flat cover FC(M ) is also cotorsion. Suppose that a simple functor A ⊆ − R M on Flat-R is generated by an element a ∈ A(U ), with U an indecomposable flat cotorsion module. There is an S-morphism ζ : U R MS → IS such that ζ(a) = 0. By the adjoint property, there is a corresponding morphism of right R-modules ζ : UR → M with the property that the following diagram commutes. U

R

ζ ⊗ 1M -

M

M

R

M

@

@

@ζ @ @ R @

ν

? IS

Here ν : M R M → IS is the canonical morphism determined by f ⊗ m → f (m). This shows that (ζ ⊗ 1M )(a) = 0.

Proposition 4. Let M be a left R-module and c : FR → M the flat cover of M . If A ⊆ − R M is a simple functor on Flat-R, then A(F ) = 0. Therefore if UR is the indecomposable flat cotorsion module for which A(U ) = 0, then UR occurs as a direct summand of the flat cover of M . Proof. Let c : F → M be the flat cover. There is a morphism g : U → F such that the following diagram commutes.


801

U

g

ζ

?

c

F Applying the functor − S-modules.

R

- M

M we obtain the following commutative diagram of right U ⊗M

ζ ⊗ 1M

g ⊗ 1M

F

M R

c ⊗ 1M -

M

? R

M

Because (ζ ⊗ 1M )(a) = 0, we see that (g ⊗ 1M )(a) = 0. In particular, A(g)(a) = 0 and so A(g) : A(U ) → A(F ) is non-zero. Let A ⊆ − R M be a simple functor on Flat-R. By Zorn’s lemma, there is a submodule N ⊆ M maximal with respect to the property that A − ⊗ N . The simple functor A is then isomorphic to a subfunctor of − R (M/N ) ∼ = (− ⊗ M )/(− ⊗ N ). The module M/N is uniform, so its injective envelope E = E(M/N ) is indecomposable. However, − ⊗ M/N is isomorphic to a subfunctor of − R E. We conclude that every simple functor on Flat-R which is a subfunctor of a continuous functor is isomorphic to a subfunctor of a functor of the form − ⊗ E, where R E is an indecomposable injective left R-module. 2. The socle of a continuous functor By [4, Proposition 2], there exists a set of flat right R-modules {F i : i ∈ I} such that CR is cotorsion if and only if Ext1R (Fi , C) = 0 for every i ∈ I. Letting T = i Fi yields a flat module that serves as a test module for the cotorsion property in the sense that Ext1R (T, C) = 0 if and only if CR is cotorsion. We may assume that |T | ℵ. If MR is a right R-module such that |M | |T | and λ is a cardinal such that λ|T | = λ, then we can apply [8, Theorem 2] with B = T , L = M , κ = |T | and λ = 2|T | to obtain a cotorsion module AR ⊇ MR of cardinality λ, such that the quotient A/M is flat. The following is then immediate. Proposition 5. Let MR be an R-module such that |M | ℵ. Then the cotorsion envelope CE(M ) of M has cardinality at most 2|T | . If MR is contained in a flat cotorsion module M ⊆ CR , then it is contained in a direct summand of C of cardinality at most 2|T | . Proof. The first statement follows from the preceding remarks. For the second, apply the Löwenheim–Skolem theorem (see, for example, [17, Theorem 5.1.1]) to obtain an elementary submodule F ⊆ CR of cardinality at most ℵ, containing MR . Then FR is a flat pure submodule of CR , so its cotorsion envelope CE(F ) is isomorphic over FR to a direct summand of C. By the first statement, the cardinality of CE(F ) is at most 2|T | .

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Proposition 5 implies that the cardinality of an indecomposable flat cotorsion module UR is bounded by 2|T | . Just let MR = aR be any non-zero cyclic submodule in the second part of the proposition. It follows that, up to isomorphism, the indecomposable flat cotorsion modules form a set. Lemma 6. Let M be a left R-module. Every non-zero functor B : C → Ab which is a subfunctor of (− ⊗ M )|C contains a non-zero subfunctor Ba ⊆ B finitely generated on C by a ∈ B(C) with CR ∈ C, a cotorsion module of cardinality at most 2|T | . Proof. By Lemma 2, there is a subfunctor Ba ⊆ B finitely generated on C by a ∈ B(C) for some cotorsion module CR ∈ C. Since a ∈ B(C) ⊆ C R M , we may write it as a = i ci ⊗ mi . Proposition 5 implies that the ci are contained in a direct summand C0 ⊆ C of cardinality C0 a retraction of i, then at most 2|T | . If i : C0 → C is the inclusion morphism and p : C → a0 = (p ⊗ 1M )(a) is just the element a viewed as a member of C0 R M . We also have a = (i ⊗ 1M )(a0 ), showing that a ∈ C ⊗ M and a0 ∈ C0 ⊗ M generate the same subfunctor. Since the category C is closed under summands, the module C0 also belongs to C. Theorem 7. Let C ⊆ Flat-R be a full additive subcategory closed under direct limits and cotorsion envelopes, and M a left R-module. Every functor B : C → Ab which is a subfunctor of (− ⊗ M )|C contains a simple subfunctor. Proof. Suppose that B does not contain a simple subfunctor. We will produce a strictly descending chain B ⊇ B0 ⊇ B1 ⊇ . . . ⊇ Bα ⊇ . . . of subfunctors, indexed by the ordinals, having the property that for every ordinal α, Bα is a finitely generated functor on C, with a generator cα ∈ B(Cα ), where Cα is a cotorsion module in C of cardinality at most 2|T | . Then some cotorsion module C ∈ C will appear as Cα more than |B(Cα )| times, contradicting the strictness of the chain. Let us build a totally ordered directed system (Cα , fαβ ), where each Cα is a cotorsion module in C of cardinality at most 2|T | , with the further property that there is an element c0 ∈ B(C0 ) such that the elements cα = B(f0α )(c0 ) generate the subfunctors Bα ⊆ B of the strictly descending chain. (i) For α = 0, take any cotorsion module C0 ∈ C of cardinality at most 2|T | with a non-zero c0 ∈ B(C). As B is non-zero, Lemma 6 ensures the existence of such a C0 ∈ C and c0 ∈ C0 . (ii) Suppose that the directed system has been defined for α γ such that the corresponding subfunctor Bγ is non-zero. If B does not contain a simple subfunctor, then neither does Bγ . Take a proper non-zero subfunctor Bγ ⊆ Bγ and proceed as in the initial stage. Let Cγ+1 be any cotorsion module in C of cardinality at most 2|T | for which there is a non-zero element cγ+1 ∈ Bγ (Cγ+1 ). Once again, Lemma 6 guarantees the existence of such a module and element, and because Bγ is a proper subfunctor of Bγ , the subfunctor Bγ+1 , defined to be the subfunctor of B generated by cγ+1 ∈ B(Cγ+1 ), will also be a proper subfunctor of Bγ . Moreover, cγ+1 belongs to the subfunctor of B generated by cγ , so there is a morphism fγ,γ+1 : Cγ → Cγ+1 such that B(fγ,γ+1 )(cγ ) = cγ+1 . (iii) Suppose finally that γ is a limit ordinal, and that the directed system has been defined for α < γ. The direct limit Fγ = lim Cα with structural morphisms fαγ : Cα → Fγ belongs to → )(cα ) C. By the continuity of − ⊗ M and Lemma 1, the subfunctor Bγ generated by cγ := B(fαγ (for any α) is non-zero and properly contained in all the subfunctors Bα , for α < γ. By Lemma 6, there is a non-zero finitely generated subfunctor Bγ ⊆ Bγ generated by an element cγ ∈ B(Cγ ) with Cγ , a cotorsion module in C, of cardinality at most 2|T | . There is a morphism . gγ : Fγ → Cγ such that cγ = B(gγ )(cγ ). Let fαγ = gγ fαγ


Applying Theorem 7 with C = Flat-R, M = R, and B = − consequence. Corollary 8. module UR .

R

803

R yields the following

Every associative ring R has a non-zero indecomposable flat cotorsion

Theorem 9. Let C ⊆ Flat-R be a full additive subcategory closed under direct limits and cotorsion envelopes. Every module FR ∈ C admits a pure monomorphism into a direct product U of indecomposable cotorsion modules Ui ∈ C. i i Proof. Let R N be a finitely presented left R-module. The restriction to C of the continuous functor − R N is continuous; so if a ∈ F R N , then Theorem 7 implies that the subfunctor Ba of (− ⊗ N )|C finitely generated on C by a ∈ F ⊗ N contains a simple subfunctor AN,a ⊆ Ba . Let UN,a be the indecomposable cotorsion module in C for which AN,a (UN,a ) = 0. There is a morphism f : F → UN,a such that (f ⊗ 1N )(a) = 0. Takingthe product of the UN,a , as N ranges over the finitely presented left R-modules and a ∈ F R N , yields a monomorphism μ of FR into a direct product of indecomposable cotorsion modules from C. Finally, note that μ must be pure since, by construction, for any finitely presented N , and any left R-module element a ∈ F ⊗ N , the (N, α)th component of μ(a) in F ⊗ U(N,α) ∼ = (F ⊗ U(N,α) ) is non-zero. Notice that while a product of cotorsion modules is always cotorsion, a product of flat modules need not be flat. In the case when C = Flat-R, the statement of Theorem 9 is given as follows. Corollary 10. Every flat module FR admits a pure monomorphism into a product of indecomposable flat cotorsion modules. 3. The cotorsion envelope of a ring We begin this section by recalling the following result from [11, Lemma 11]. Proposition 11. A morphism m : PR → FR from a projective module PR to a flat module FRis a pure monomorphism if for every simple left R-module R X, the morphism m ⊗ 1X : P R X → F R X is a monomorphism of abelian groups. Let us define P ⊆ Flat-R to be the class of all flat right R-modules that enjoy this property of projective modules. In other words, a flat right module G belongs to P if: whenever a morphism m : G → F has the property that m ⊗ 1X is a monomorphism for all simple left R-modules X, then m is a pure monomorphism. The following implies that every summand of the cotorsion envelope of RR belongs to the class P, since the embedding of any module in its cotorsion envelope is strongly pure-essential by [11, p. 14]. Proposition 12. The class P ⊆ Flat-R is closed under strongly pure-essential extensions and direct summands. Proof. Let F ∈ P and suppose that F ⊆ F is a strongly pure-essential extension. If m : F → G is a morphism to a flat module G such that m ⊗ 1X is a monomorphism for every simple R X then, as F is a pure submodule of F , the same is true of the restriction m|F . By hypothesis, m|F : F → G is a pure monomorphism. As F ⊆ F is a strongly pure-essential extension, m : F → G is also a pure monomorphism.

804


Now suppose that F = F ⊕ F belongs to P. If m : F → G is a morphism to a flat module G such that m ⊗ 1X is a monomorphism for every simple module X, then the morphism m 0 : F ⊕ F → G ⊕ F 0 1F has the same property. By hypothesis, it is pure. Thus m : F → G is also pure. We do not know whether P is closed under pure submodules. Proposition 13. Let G ∈ P, and let F ⊆ G be a non-zero pure submodule. Then there is a simple left R-module X such that F R X = 0. p i Proof. Consider the pure-exact sequence 0 → F → G → G/F → 0. If F R X = 0 for every simple left R-module X, then p ⊗ 1X : G R X → G/F R X is a monomorphism. As G belongs to P, the morphism p : G → G/F must then be a pure monomorphism, and hence a monomorphism. However, F = 0, a contradiction. For example, if G ∈ P and e ∈ S = EndR G is a non-zero idempotent, then there must be a simple X such that the morphism e ⊗ 1X is non-zero. This is because e ⊗ 1X : left R-module G R X → eG R X is a surjection, and eG is a pure submodule of G. The following is a slight generalization of [11, Proposition 12]. Proposition 14. Let F ∈ P be cotorsion, F ⊆ C the cotorsion envelope, and S = EndR C. If f ∈ S satisfies f |F ⊗ 1X = 0 for every simple left R-module X, then f ∈ J(S). Proof. If f ∈ J(S), then by the comments preceding [10, Theorem 6], there is a non-zero idempotent e ∈ Sf . If f |F ⊗ 1X = 0 for every simple module X, then e|F ⊗ 1X = 0. It follows that (1C − e)|F ⊗ 1X = 1|F ⊗ 1X is a monomorphism for every simple X. Because F ∈ P, the morphism (1C − e)|F is a pure monomorphism. As C is a strongly pure-essential extension of F , the morphism 1C − e : C → C must also be a pure monomorphism. However, 1C − e is a monomorphism, so it must be 1C , which implies that e = 0, a contradiction. The argument just used in the proof of Proposition 14 will appear again. Let us state it for later reference. Lemma 15. Let g : F → CE(F ) be the cotorsion envelope of a flat module FR . If e : CE(F ) → CE(F ) is an idempotent such that the restriction eg : F → CE(F ) is a pure monomorphism, then e = 1CE(F ) . Let C = CE(R) be the cotorsion envelope of the module RR and denote by S = EndR C the endomorphism ring of C. If c ∈ C, then there is a unique R-morphism fc : R → C satisfying fc (1) = c. By the definition of cotorsion envelope, the morphism fc extends to a morphism fc : C → C. The next proposition shows that any two such endomorphisms differ by an element of J(S). Proposition 16. If C = CE(R) and S = EndR C, then annS (1) ⊆ J(S). The evaluation morphism Ev1 : f → f (1) therefore induces an isomorphism Ev1 : S/J(S) → C/rad(C) of left S-modules.


805

Proof. If f ∈ S belongs to annS (1), then f (R) = 0, so that for any g ∈ S, we have gf (R) = 0. Thus, the element 1 − gf acts as the identity on R. By the definition of cotorsion envelope, it is necessarily an invertible element of S. Thus f ∈ J(S). The evaluation morphism Ev1 : S S → S C is an epimorphism of left S-modules, so it induces an isomorphism C ∼ = S/annS (1) of cyclic S-modules. An isomorphism from S/J(S) to C/rad(C) is induced because every maximal left ideal of S contains annS (1). If r ∈ R ⊆ C, then the rule r → fr induces a ring morphism ϕ : R → S/J(S). This follows from the equation fr fs (1) = fr (s) = fr (1)s = rs = frs (1), for any choice of fr , fs and frs . Recall from [5] that a morphism of rings is local if any preimage of a unit is a unit. Theorem 17. Let C = CE(R) be the cotorsion envelope of RR , and let S = EndR C. The ring morphism ϕ induces a local monomorphism of rings denoted by ϕ : R/J(R) → S/J(S). Proof. First, let us show that J(R) = Ker ϕ. If r ∈ J(R), then for every simple left R-module X, rX = 0, so we see that fr ⊗ 1X = 0. By Proposition 14, with F = R, fr ∈ J(S). If, on the other hand, r ∈ J(R), then there is a simple left R-module X such that rX = 0. Choose x ∈ X such that rx = 0. As X is simple, there is an element s ∈ R such that srx = x or (1 − sr)x = 0. This implies that (1 − fs fr )|R ⊗ 1X = (1 − fs fr ) ⊗ 1X is not an isomorphism. However, as R is pure in C = CE(R), neither is (1 − fs fr ) ⊗ 1X an isomorphism. However, 1 − fs fr cannot be a unit in S, and therefore fr ∈ J(S). To see that ϕ is a local morphism of rings, suppose that ϕ(r) is a unit in S/J(S). Then any representative fr ∈ S is an automorphism. In particular, the restriction fr = fr |R : RR → RR is a pure monomorphism. The cokernel of fr is finitely presented, so it is pure-projective. It follows that fr is a split monomorphism and, therefore, that rR = eR for some idempotent e ∈ R. Now (1 − e)r = 0 implies that (1 − ϕ(e))ϕ(r) = 0, and ϕ(r) is a unit. Thus 1 − ϕ(e) = 0, which implies that 1 − e ∈ J(R), by the first part of the theorem. However, e = 1 and rR = R so that r has a right inverse s; rs = 1. Since ϕ(r) is invertible and ϕ(r)ϕ(s) = 1, the inverse of ϕ(r) is ϕ(s). In R, the product sr = k ∈ R is idempotent; (sr)(sr) = s(rs)r = sr. By the first part of the theorem 1 − k ∈ J(R), which implies that k = 1. Corollary 18. Let R be an associative ring with identity. Then R is local if and only if CE(RR ) is indecomposable. Proof. If R is local, then CE(RR ) is indecomposable by [11, Corollary 16]. Conversely, if C = CE(RR ) is indecomposable then S = EndR CR is a local ring [10, Corollary 7] and so S/J(S) is a division ring. By Theorem 17, every non-zero element of R/J is a unit, so that R/J is also a division ring, which implies that R is a local ring with maximal ideal J. Remark 19. Let R be an arbitrary ring. We can associate to R the ring S = End(CE(RR )). Theorem 17 and Corollary 18 show that the homomorphism of rings φ : R/J(R) → S/J(S) is a local homomorphism, and thus R is semilocal if and only if S is. As S is the endomorphism ring of a flat cotorsion module, S/J(S) is von Neumann regular and right self-injective, and the idempotents lift modulo J(S) (see [10]). This means that R is semilocal if and only if S is semiperfect. In other words, our results show how to associate a semiperfect ring to any semilocal ring in a very natural way. It is well known that finitely generated projective modules over a semiperfect ring are just the projective covers of finitely generated semisimple modules, which therefore determine their structure. However, finitely generated projective modules over the original semilocal ring R

806


need not be the covers of finitely generated semisimple R-modules. This is the main reason for the pathologies of projective modules over semilocal rings (see, for example, [9]). Our aim in the next section is to show that the natural class associated to finitely generated semisimple modules over a semilocal ring R is not the class of finitely generated projective modules, but the class of direct summands of finite direct sums of copies of CE(RR ). The reason is that, when the ring morphism ϕ : R/J → S/J(S) is composed of the isomorphism Ev1 : S/J(S) → C/rad(C) of Proposition 16, one obtains a morphism of right R-modules that fits into the following commutative diagram of right R-modules. R

- CE(R)

ε

? R/J

? Ev1 ◦ ϕ -

C/rad(C)

Here ε : R → CE(R) is the cotorsion envelope of RR , and both vertical arrows represent natural quotient maps. In particular, we can relate the structure of indecomposable direct summands of CE(RR ) to the structure of indecomposable projective S-modules over the semiperfect ring S = End(CE(RR )).

4. Semilocal rings Throughout this section, we will assume that R is a semilocal ring. Several simplifications come into effect. By [11, Proposition 12], the radical rad(C) of the cotorsion envelope C = CE(R), viewed as a left module over S = EndR C, is equal to CJ. Also, Proposition 11 takes on the following form. Lemma 20. Let R be a semilocal ring and GR a flat module that belongs to the class P. A morphism m : GR → FR to a flat module FR is a pure monomorphism if and only if the induced morphism of right R/J-modules m/J : G/GJ → F/F J is a monomorphism. Let us fix some notation. Let X1 , X2 , . . . , Xm be a complete list, without repetition, of the simple right R-modules, up to isomorphism, and let Ji = annR (Xi ). The length of R/Ji , which

is isomorphic to the Xi -homogeneous component of R/J, will be denoted by ni . Thus J = i Ji and the length of m (n ) R/J = Xi i is given by n =

i

i=1

ni .

Theorem 21. Let R be a semilocal ring. There is a decomposition of the cotorsion envelope as a direct sum m (n ) CE(RR ) = Ci i i=1

of indecomposable modules with the property that Ci /Ci J ∼ = Xi .

807


Proof. For ease of notation, let us write R = R/J, S = S/J(S) and denote the cotorsion envelope ε : RR → CE(R) by C. Since R=

m

(ni )

Xi

,

i=1

there is a maximal collection of pairwise orthogonal primitive idempotents {œij }ij in R such that œij R ∼ = Xi , and ni m R= œij R. i=1 j=1

Thus {ϕ(œij )}ij is a collection of pairwise orthogonal elements of S that partition unity: ⎞ ⎛ ni ni m m ϕ(œij ) = ϕ ⎝ œij ⎠ = ϕ(1R ) = 1S . i

j

i

j

a collection The collection {ϕ(œij )}ij may be lifted modulo J(S) by [10, Corollary 4] to {eij }ij of pairwise orthogonal idempotent elements of S with the property that ij eij = 1. Let Cij := eij C. Then ni m Cij C = CE(R) = i=1 j=1

and C/CJ = ij Cij /Cij J. Considering the left square in the commutative diagram R

ε

- CE(R)

? R

1 − ekl + ekl

-

? Ev1 ◦ ϕ -

C/CJ

C

? 1 − ϕ(œkl ) + œkl

-

C/CJ

we have the equation (Ev1 ◦ ϕ)(œij ) = Ev1 (ϕ(œij )) = eij 1 + CJ,

(4.1)

where 1 ∈ C refers to ε(1). Thus (Ev1 ◦ ϕ)(œij R) ⊆ Cij /Cij J. Suppose that one of the Cij , say Ckl , is not indecomposable, Ckl = Ckl ⊕ Ckl . According to this direct sum decomposition, we may decompose ekl = ekl + ekl into a non-trivial sum of orthogonal idempotents in S. Since œkl ∈ R is non-zero, Equation 4.1 implies that ekl 1 ∈ CJ. It must be that either ekl 1 ∈ Ckl J or that ekl 1 ∈ Ckl J. We may suppose, without loss of generality, that the first one holds. Denote by œkl the idempotent endomorphism induced on C/CJ by ekl . We will show that the composition of the two morphisms in the bottom row of the commutative diagram is a monomorphism. Then Proposition 11 implies that (1 − ekl + ekl )ε is a pure monomorphism and Lemma 15 yields the contradiction 1 − ekl + ekl = 1. Since 1 − ekl and ekl are orthogonal, so are 1 − ϕ(œkl ) and œkl . However, (1 − ϕ(œkl ))(Ev1 ◦ ϕ)(œij R) ⊆ Cij /Cij J is non-zero for all (i, j) = (k, l), and œkl (ekl R) ⊆ Ckl /Ckl J

808


is also non-zero. The image of (1 − ϕ(œkl ) + œkl )(Ev1 ◦ ϕ) therefore has the same finite length n as R. It must be that this composition of morphisms is a monomorphism. We have established that the cotorsion envelope CE(R) is a direct sum of the n = i ni indecomposable modules Cij . Each Cij is an indecomposable flat cotorsion module, so it has a local endomorphism ring. By [2, Theorem 12.6] and [7, Theorem 3.4], this decomposition complements direct summands and, by the Krull–Schmidt theorem (see [2, Theorem 12.9]), it is essentially unique. For each i, 1 i m, the R-module Ci1 /Ci1 J contains (Ev1 ◦ ϕ)(œij R), an R-module (n ) (n ) isomorphic to Xi . Therefore if we let C = i Ci1 i , then we may embed R = i Xi i into the quotient module C /C J. The lift q of this embedding to R is represented by the vertical arrow in the following commutative diagram.

C

ppp ppp

εp p p

pp ppp

pp ppp

ppp

p

R

p ppp

q

? - C /C J

By Proposition 20, the morphism ε : R → C is a pure monomorphism, which extends to a pure monomorphism ε : CE(R) → C on the cotorsion envelope. This implies that CE(R) m (n ) is a direct summand of C = i=1 Ci1 i . However, this decomposition of C complements direct summands so that no proper summand can decompose as a direct sum of n modules. The morphism ε : CE(R) → C is therefore onto. Abbreviating Ci1 to Ci , one obtains a decomposition (n ) CE(R) = Ci i , i

where each Ci /Ci J contains a submodule isomorphic to Xi . Suppose that there is a simple right R-module Xk that occurs as a submodule of some Ci /Ci J, where i = k. Consider the module (n ) (n −1) C = Ci i ⊕ Ck k ; i=k

it has the property that C /C J contains a module isomorphic to R. An argument as in the previous paragraph yields a direct summand isomorphic to CE(R). Again, the given decomposition of C as a direct sum of n − 1 modules complements direct summands so that C cannot have a direct summand that is a direct sum of n non-zero modules. We conclude that each of the modules Ci /Ci J is a homogeneous semisimple module. Finally, let us show that each Ci /Ci J is simple. Suppose that there is a k, 1 k m, such (α) that Ck /Ck J ∼ = Xk with α > 1. Consider the module (2n ) (n ) Ci i ⊕ Ck k . C = i=k

The quotient C /C J contains a submodule isomorphic to R ⊕ R. As above, this yields a direct summand of C isomorphic to the cotorsion envelope CE(R ⊕ R) ∼ = CE(R) ⊕ CE(R). The given decomposition of C as a direct sum of 2n − nk modules complements direct summands, so that no direct summand can be a direct sum of 2n non-zero modules.


809

Theorem 21 implies that the cotorsion envelope ε : RR → CE(R) induces an isomorphism of R/J-modules, ε/J : R/J → C/CJ. Thus the ring monomorphism ϕ : R/J → S/J(S) given in Theorem 17 is an isomorphism. As R is a semilocal ring, every simple right R-module Xi is finite-dimensional as a module over its endomorphism ring Δi . The local dual operation Xi → Xi yields a bijection (in either direction) between the isomorphism types of simple right R-modules and those of simple left R-modules. Furthermore, the endomorphism ring of Xi is isomorphic to Δi and the canonical map from Xi R Xi → Δi is an isomorphism of Δi –Δi -bimodules. Corollary 22. If A : Flat-R → Ab is a simple subfunctor of the continuous functor − ⊗ Xi , then A(Ci ) = 0. Proof. By Theorem 3, there is an indecomposable flat cotorsion module UR such that A(U ) ⊆ U ⊗ Xi is non-zero. Choose a non-zero a ∈ A(U ). Because Xi is simple, the element a = u ⊗ x, say, of U ⊗ Xi is an elementary tensor. Also annR (Xi ) = annR (Xi ) = Ji , so that u ∈ U Ji . There is a morphism fu : RR → U given by fu (1) → u. Since UR is cotorsion, the morphism fu may be extended to a morphism fu : CE(RR ) → U . Now the image Imfu is not contained in U Ji , and by Theorem 21, Ck Ji = Ck for all k = i. Therefore, it must be that the restriction of fu to one of the summands Ci is not contained in U Ji , fu (Ci ) U Ji . By Lemma 20 and the fact that Ci /Ci Ji is simple, the restriction of fu to Ci is a pure monomorphism. It is therefore a split monomorphism and, since U is indecomposable, it must be isomorphic to Ci . Corollary 23. Let R be a semilocal ring and Ci an indecomposable summand of the cotorsion envelope CE(RR ). The quotient morphism qi : Ci → Ci /Ci J = Xi is the flat cover of Xi . Proof. Consider the flat cover gi : FC(Xi ) → Xi . We will show that Ker gi = FC(Xi )J. Proposition 4 and Corollary 22 imply that Ci is a direct summand of FC(Xi ): FC(Xi ) = Ci ⊕ C . If Ker gi = FC(Xi )J, then FC(Xi )/FC(Xi )J = Ci /Ci J ⊕ C /C J is a simple module isomorphic to Xi . This forces the summand C = C J ⊆ FC(Xi )J to be contained in the kernel of gi . Because a flat cover contains no non-zero summands in the kernel, C = 0. Evidently, FC(Xi )J ⊆ Ker gi . Hence suppose that this inclusion is proper, and choose an element a ∈ Ker gi \ FC(Xi )J. Because Ker gi is a cotorsion module, the morphism fa : RR → Ker gi given by fa (1) → a may be extended to a morphism fa : CE(RR ) → Ker gi as in the following commutative diagram. CE(R) 6@ ε

ιfa

R

6

@ @ f a

- FC(Xi )

ι

@

fa

@ R @ - Ker gi

810


Here ι : Ker gi → FC(Xi ) is the inclusion morphism. The image of ιfa : CE(R) → FC(Xi ) is not contained in FC(Xi )J, so the same must be true of the restriction to one of the indecomposable summands Ck of CE(RR ). By Lemma 20 and the observation that Ck /Ck J is simple, the restriction to Ck of ιfa is a pure monomorphism, which is necessarily split, because FC(Xi ) is flat. Then the restriction of fa to Ci is a split monomorphism and the Ker gi contains a non-zero direct summand of FC(Xi ), a contradiction. Corollary 23 implies that the quotient morphism q : CE(R) → CE(R)/CE(R)J ∼ = R/J is the flat cover of the right R-module R/J, and therefore that the module CE(R)J is cotorsion. Consider the short exact sequence - Ci Ji

0

ιi

- Ci

qi

- Xi

-

0,

where qi : Ci → Xi is the flat cover of Xi . Applying the functor − ⊗ Xi yields an exact sequence of abelian groups Ci Ji ⊗ Xi

0

- Ci ⊗ X

- Xi ⊗ X

i

i

-

0.

The morphism ιi ⊗ 1X : Ci Ji ⊗ Xi → Ci ⊗ Xi i

is zero, because Ji = Thus Ci ⊗ Xi ∼ = Xi ⊗ Xi . The proof of the next result shows that if M is a simple left R-module, then the functor − ⊗ M restricted to the category of flat cotorsion right R-modules is simple.

annR (Xi ).

Corollary 24. If M is a simple left R-module, then the continuous functor − ⊗ M : Flat-R → Ab contains a unique simple subfunctor. Proof. There is a simple right R-module Xi such that M ∼ = Xi . Hence if A ⊆ − ⊗ M is a simple subfunctor, then by Corollary 22, the abelian group A(Ci ) ⊆ Ci ⊗ X ∼ = Xi ⊗ X i

i

is non-zero. Now every endomorphism of Xi is induced by an endomorphism from its flat cover Ci , and the endomorphisms of Ci act on A(Ci ). Since Xi ⊗ Xi is a one-dimensional left vector space over the endomorphism ring Δi , it must be that A(Ci ) = Ci ⊗ Xi . If B ⊆ − ⊗ Xi were another simple functor, then by Corollary 22, B(Ci ) would also be non-zero, and hence we would have B(Ci ) = Ci ⊗ Xi = A(Ci ). However, two simple functors that contain each other’s generators must be equal. References ´mek and J. Rosicky ´ , Locally presentable and accessible categories, London Mathematical Society 1. J. Ada Lecture Note Series 189 (Cambridge University Press, Cambridge, 1994). 2. F. W. Anderson and K. R. Fuller, Rings and categories of modules, Graduate Texts in Mathematics 13 (Springer, London 1992). 3. M. Auslander, ‘Functors and morphisms determined by objects’, Representation theory of algebras, Proceedings of the conference at Temple University, 1976, Lecture Notes in Pure and Applied Mathematics 37 (Dekker, New York, 1978) 1–244. 4. L. Bican, R. El Bashir and E. Enochs, ‘All modules have flat covers’, Bull. London Math. Soc. 33 (2001) 385–390. 5. R. Camps and W. Dicks, ‘On semilocal rings’, Israel J. Math. 81 (1993) 203–211. 6. W. W. Crawley-Boevey, ‘Locally finitely presented additive categories’, Comm. Algebra 22 (1994) 1641–1674. 7. N. V. Dung, ‘Modules with indecomposable decompositions that complement maximal direct summands’, J. Algebra 197 (1997) 449–467.

INDECOMPOSABLE FLAT COTORSION MODULES 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

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C. Eklof and J. Trlifaj, ‘How to make Ext vanish’, Bull. London Math. Soc. 33 (2001) 41–51. Facchini, ‘Module theory’, Progress in Mathematics 167 (Birkh¨ auser, Boston, MA, 1998). A. Guil Asensio and I. Herzog, ‘Left cotorsion rings’, Bull. London Math. Soc. 36 (2004) 303–309. A. Guil Asensio and I. Herzog, ‘Sigma-cotorsion modules and divisibility matrix subgroups’, Adv. Math. 191 (2005) 11–28. P. A. Guil Asensio and I. Herzog, ‘Model-theoretic aspects of Σ-cotorsion modules’, Ann. Pure Appl. Logic 146 (2007) 1–12. D. K. Harrison, ‘Infinite abelian groups and homological methods’, Ann. of Math. 69 (1959) 366–391. I. Herzog, ‘Pure-injective envelopes’, J. Algebra Appl. 2 (2003) 397–402. ´, Accessible categories: the foundations of categorical model theory, Contemporary M. Makkai and R. Pare Mathematics 104 (American Mathematical Society, Providence, RI, 1989). M. Prest, Model theory of modules, London Mathematical Society Lecture Note Series 130 (Cambridge University Press, Cambridge, 1998). P. Rothmaler, Introduction to model theory, Algebra, Logic and Applications Series 15 (Gordon and Breach Science Publishers, Amsterdam, 2000). J. Xu, Flat covers of modules, Lecture Notes in Mathematics 1634 (Springer, New York, 1996). M. Ziegler, ‘Model theory of modules’, Ann. Pure Appl. Logic 26 (1984) 149–213.

P. A. P. P.

Pedro A. Guil Asensio Departamento de Matemáticas Universidad de Murcia 30100 Espinardo Murcia Spain paguil@um·es

Ivo Herzog The Ohio State University at Lima Lima OH 45804 USA herzog·23@osu·edu