FUNCTORIAL PROPERTIES OF HOM AND EXT 1. Introduction Every ...

FUNCTORIAL PROPERTIES OF HOM AND EXT ULRICH ALBRECHT, SIMION BREAZ AND PHILL SCHULTZ

Abstract. The goal of this paper is to present a comprehensive survey of recent results addressing the question for which Abelian groups G the functors Hom(G, −), Hom(−, G), Ext(G, −) or Ext(−, G) preserve or invert direct products or sums.

1. Introduction Every right R-module A induces three families of functors, namely the covariant functors A ⊗R − and HomR (A, −) together with their induced functors TorR n (A, −) and ExtnR (A, −), and the contravariant functor HomR (−, A) with its derived functors ExtnR (−, A). Being additive, each of these functors commutes with finite direct sums. Although their behavior with respect to infinite sums and products has given rise to many interesting classes of modules, a comprehensive survey of this important topic has so far been absent from the literature. It is the goal of this paper to present such a survey with a special emphasis on the functors Hom and Ext in the context of Abelian groups. Although we are mostly interested in the functors Hom and Ext, we begin our discussion in a more general setting by considering an additive functor F = FA which is induced by a bifunctor. While any reference to A is usually suppressed, we nevertheless want to mention that the maps constructed in the following are natural with respect to A. For a class C of R-modules, choose modules {Ni }i∈I from C and consider the modules F(⊕I Ni ), ⊕I F(Ni ), F(ΠI Ni ), and ΠI F(Ni ). The universal properties of direct sums and products induce natural homomorphisms between these objects. In Section 2, we define whether F preserves or inverts direct sums (products) with respect to the class C by specifying which of these maps are isomorphisms (Proposition 2.1 and Proposition 2.4). If no reference to C is given, then C = MR , the class of all right R–modules. Since F = FA , the class C = {A} merits special attention. In this case, we say that F preserves (inverts) self–sums (self–products). It is well-known that HomR (A, −) and ExtnR (A, −) preserve direct products since the induced maps ExtnR (A, ΠI Ni ) → ΠI ExtnR (A, Ni ) are natural isomorphisms for all n < ω [CE56]. Similarly, the contravariant functors HomR (−, A) and ExtnR (−, A) invert direct sums. However, if a functor F already has one of these two properties, then the requirement that it has an additional preservation Date: October 3, 2011. 1991 Mathematics Subject Classification. 20K35, 20K40. Key words and phrases. Functors on Abelian groups, extensions, self–sums, self–products, torsion–free Abelian groups. S. Breaz is supported by the CNCS- UEFISCDI grant PN-II-RU-TE-2011-3-0065 . 1

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or inversion property for families from a class C often yields F(C) = 0. Nevertheless, the cases when F does not operate trivially on C give rise to some important classes C of modules, for instance the slender modules, i.e. those modules A for which HomR (−, A) inverts products, and the (self–) small modules, i.e., those for which HomR (A, −) preserves sums (of copies of A). Hence, HomR (A, −) is a covariant functor which preserves direct sums and direct products whenever A is small. While every finitely generated module is small, there are rings for which one can find small modules which are not finitely generated [EGT97]. Although this does not occur in the case of Abelian groups (Corollary 4.2), the mixed self-small groups are one of the most widely studied classes of Abelian groups. Section 4 presents further properties of these groups continuing the investigation of self-small groups started in [AM75], [AGW95] and [ABW09]. Finally, the ideas of Proposition 2.1 are not limited to Hom and Ext functors. Using the universal properties of tensor products it is easy to see that A ⊗R − preserves direct sums. However, the functor A ⊗R − preserves direct products if and only if the right R-module A is finitely presented [Go76]. In the remaining part of the paper, we apply these general results to the special case where MR is the category A of Abelian groups and F is one of the covariant functors Hom(A, −) or Ext(A, −) or one of the contravariant functors Hom(−, A) or Ext(−, A). The main results of Section 3 give necessary and sufficient conditions on Abelian groups A for Hom(A, −) or Ext(A, −) to invert sums and for Hom(−, A) or Ext(−, A) to preserve products from some classes C. In particular, Hom(−, A) inverts all products from A if and only if A is strongly slender. However, there are cases in which Hom(−, A) inverts products from a class C without A being slender. The investigation of the corresponding properties for Ext(−, A) leads to the new class C–coextendible groups whose study is similar to the study of cotorsion theories and their generalizations [Sa79], [Str02], [StrW01]. Section 5 investigates C-extendible groups, which are the groups A for which Ext(A, −) preserves direct sums. We show that the structure of extendible and self-extendible groups is trivial in the sense that a group is extendible if and only if it is a direct sum of a finitely generated group and a free group. We also prove a similar structure theorem for self-extendible groups (Theorem 5.3). To conclude this introduction, we want to mention another interesting line of inquiry, which we cannot consider within the framework of this paper: for which modules (Abelian groups) are the natural maps discussed in Section 2 either monomorphisms or epimorphisms? Obtaining answers to these questions would be particularly interesting for the Ext-functors. For tensor products, the former question yields the interesting class of Mittag-Leffler modules, while the latter leads to the class of finitely generated modules.

2. Functors Acting on Sums and Products Let C be a non–empty class of R-modules; and consider a non-empty family N = {Ni }I of modules from C. The symbol fi denotes the canonical embedding of Ni into the ith -coordinate of ⊕j∈I Nj , while fi0 indicates the corresponding embedding into Πj∈I Nj . Similarly, the symbols gi : ⊕j∈I Nj → Ni and gi0 : Πj∈I Nj → Ni denote the canonical projections onto the ith -coordinate. Finally, ιN : ⊕j∈I Nj → Πj∈I Nj is the canonical embedding.

FUNCTORIAL PROPERTIES OF HOM AND EXT

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We begin our discussion with the case that F is a covariant Q functor defined on MR . Let φi : F(Ni ) → ⊕j∈I F(Nj ) and φ0i : F(Ni ) → j∈I F(Nj ) denote the canonical injections, while γi : ⊕j∈I F(Nj ) → F(Ni ) and γi0 : Πj∈I F(Nj ) → F(Ni ) are the canonical projections. Using the universal properties of sums and products, we obtain natural homomorphisms ΦN : ⊕I F(Ni ) → F(⊕I Ni ) and Φ0N : ⊕i∈I F(Ni ) → F(ΠI Ni ) induced by F(fi ) : F(Ni ) → F(⊕j∈I Nj ) and F(fi0 ) : F(Ni ) → F(Πj∈I Nj ). Similarly, the maps F(gi0 ) : F(Πj∈I Nj ) → F(Ni ) and F(gi ) : F(⊕j∈I Nj ) → F(Ni ) induce natural homomorphisms Y Y Y Γ0N : F( Ni ) → F(Ni ) and ΓN : F(⊕I Ni ) → F(Ni ). I

i∈I

I

Proposition 2.1. These maps fit into the diagram φi

F(Ni ) −−−−→ ⊕F(Ni ) 



ΦN y

γi

⊕F(Ni ) −−−−→ F(Ni ) 



Φ0N y

Q F (g 0 ) F (fi ) F (ιN ) F(Ni ) −−−−→ F(⊕Ni ) −−−−→ F( Ni ) −−−−i→ F(Ni )  

 

ΓN y Γ0N y

φ0

i F(Ni ) −−−− →

Q

F(Ni )

Q

γ0

F(Ni ) −−−i−→ F(Ni )

where all sums and products are taken over I. Moreover, the following hold: a) The compositions of the maps in the top and bottom rows yield the respective identity maps; and the left hand and right hand squares commute. b) i) Γ0N F(ιN ) = ΓN and F(ιN )ΦN = Φ0N . ii) ΓN ΦN = Γ0N Φ0N = ιF (N ) . In particular, the whole diagram commutes. Proof. a) is immediate from the definitions. For b) i), observe that γi0 Γ0N F(ιN ) = F(gi0 )F(ιN ) = F(gi0 ιN ) = F(gi ) = γi0 ΓN for all i ∈ I. By the universal property of direct products, Γ0N F(ιN ) = ΓN . In the same way, F(ιN )ΦN ϕi = F(ιN )F(fi ) = F(ιN fi ) = F(fi0 ) = Φ0N φi for all i. The universal property of direct sums yields F(ιN )ΦN = Φ0N . ii) is established similarly.  Using this notation, we say that the functor F a) preserves the product of N if Γ0N is an isomorphism, and preserves products from C if it preserves products of all families N from a class C, b) inverts the sum of N if ΓN is an isomorphism, and inverts sums from C if it inverts sums of all families N from C, c) preserves the sum of N if ΦN is an isomorphism, and preserves sums from C if it preserves sums of all families N from C, and finally d) inverts the product of N if Φ0N is an isomorphism, and inverts products from C if it inverts products of all families N from C.

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The question arises whether a given functor F can have more than one of these properties for an infinite family N . We will address it in a series of results, and show that there are only a few non-trivial cases where this can occur. Proposition 2.2. The following are equivalent for a covariant functor F which preserves the product of a non–empty family N of modules: a) F inverts the product of N . b) F(Ni ) = 0 for all but finitely many Ni ∈ N . Proof. If F inverts the product of N , then ιF (N ) has to be an isomorphism by the previous lemma. However, this is only possible if F(Ni ) = 0 for almost all i. The converse is obvious.  For example, HomR (A, −) and ExtnR (A, −) satisfy the hypothesis of Proposition 2.2. Q In spite of this result, there may exist an isomorphism F( i∈I Ni ) ∼ = ⊕i∈I F(Ni ) for an infinite family N = (Ni )i∈I such that F(Ni ) 6= 0 for all i. However, this isomorphism cannot be natural. For instance, we can consider F = Hom(Z, −), and I a countable = Q for all i ∈ I except for a fixed index i0 and Ni0 = Qω . i Q Q set, N ω Then F( i∈I Ni ) ∼ = i∈I Ni = Qω × Qω ∼ = Q(2 ) ∼ = Qω ⊕ Q(ω) = ⊕i∈I Ni ∼ = ⊕i∈I F(Ni ). Corollary 2.3. a) A product preserving covariant functor F inverts products from C if and only if F(N ) = 0 for all N ∈ C. b) Let F be a product preserving, covariant functor and C a maximal class such that F inverts products from C. i) If F is left exact, then C = Ker(F) and is closed under products and submodules. ii) If F is right exact, then C = Ker(F) and is closed under products and epimorphic images.  We now turn to the case that F is a contravariant functor defined on MR and N = {Ni : i ∈ I} is a non–empty family from MR . The families {F(fi )}I and {F(fi0 )}I of maps induce natural homomorphisms ∆N : F(⊕i∈I Ni ) → ΠI F(Ni ) and ∆0N : F(Πi∈I Ni ) → ΠI F(Ni ) Similarly, the families {F(gi )}I and {F(gi0 )}I induce natural homomorphisms ΨN : ⊕I F(Ni ) → F(⊕i∈I Ni ) and Ψ0N : ⊕I F(Ni ) → F(ΠI Ni ). As in the covariant case, we obtain Proposition 2.4. These maps induce the diagram φi

γi

F(Ni ) −−−−→ ⊕F(Ni )



 Ψ0N y

F (f 0 )

⊕F(Ni ) −−−−→ F(Ni ) 



ΨN y

F (ιN )

F (fi )

F(Ni ) −−−−i→ F(ΠNi ) −−−−→ F(⊕Ni ) −−−−→ F(Ni )

 

 

∆N y ∆0N y

φ0

i F(Ni ) −−−− → ΠF(Ni )

γ0

ΠF(Ni ) −−−i−→ F(Ni )

where the sums and products are taken over I. Moreover the following hold:

FUNCTORIAL PROPERTIES OF HOM AND EXT

5

a) The compositions of the maps in top and bottom rows yield the identity maps; and the left and right hand squares are commutative. b) i) ∆0N = ∆N F(ιN ) and F(ιN )Ψ0N = ΨN . ii) ∆N ΨN = ∆0N Ψ0N = ιF (N ) . In particular, the whole diagram commutes. The proof is similar to the covariant case, and will therefore be omitted.



As in the covariant case, we say that F preserves (inverts) products (sums) using the vertical maps in the diagram, e.g., F inverts the sum of N if ∆N is an isomorphism and inverts sums from C if ∆N is an isomorphism for all families N from C. As in Proposition 2.2, we obtain: Proposition 2.5. Let F be a contravariant functor which inverts the sum of a family N = {Ni : i ∈ I}. Then F preserves the sum of N if and only if F(N ) = 0 for all but finitely many N ∈ N .  For example, HomR (−, A) and ExtnR (−, A) satisfy the hypothesis of Proposition 2.5. Corollary 2.6. a) A sum inverting contravariant functor F preserves sums from C if and only if F(H) = 0 for all H ∈ C. b) Let F be a sum inverting, contravariant functor and C a maximal class such that F preserves sums from C. i) If F is left exact, then C = Ker(F) and is closed under direct sums and epimorphic images. ii) If F is right exact, then C = Ker(F) and is closed under direct sums and submodules.  Therefore it remains to consider only the following properties for the functor F: a) F is a product preserving covariant functor which preserves or inverts direct sums. b) F is a sum inverting contravariant functor which preserves or inverts direct products. 3. Hom and Ext For any Abelian group A, the symbol T (A) means the torsion subgroup of A and A = A/T (A). Given a prime p, Tp (A) denotes the p-component of A and A[p] its p-socle. Finally, A(I) = ⊕I A and AI = ΠI A for all index-sets I. To simplify our notation, sum and product always refer to direct sum and direct product. Except where explicitly stated, we adopt the notation of [F70, F73]. In particular, Jp is the group of p-adic integers and Zp the localization of Z at the prime p. We are particular interested in the case that C is one of the following classes of groups: A, all Abelian groups; T F, the torsion-free groups; T , the torsion groups; D, the divisible groups; R, the reduced groups; and A[p∞ ], the p-groups. We now discuss the properties introduced in the last section for the special case that F is induced by either Hom or Ext. Our first results investigate when the covariant functors Hom(A, −) and Ext(A, −) invert sums. Theorem 3.1. The following are equivalent for Abelian groups A and G: a) Hom(A, −) inverts sums of copies of G.

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ULRICH ALBRECHT, SIMION BREAZ AND PHILL SCHULTZ

b) Hom(A, G) = 0. ι

Proof. a) ⇒ b): Consider the exact sequence G(ω)  Gω  Gω /G(ω) . Using the first diagram of Section 2, we obtain that Hom(A, ι) is an isomorphism. Consequently, φ(A) ⊆ G(ω) for all homomorphisms φ : A → Gω . For a non-zero map α ∈ Hom(A, G), define β : A → Gω by β(x) = (αn (x))n 0, there is an exact sequence pm G/pn G  G/pn G  G/pm G. Therefore, κn = |G/pn G| for n < ω is an increasing chain of cardinals with κ0 = 0. In particular, κn is infinite for almost all n. Suppose there exists 0 < n < ω such that κn is infinite, but κn+1 > κn . Then κ2n > κn too. Consider the canonical epimorphism G/p2n G → G/pn G defined by g+p2n G 7→ g + pn G which has pn G/p2n G as its kernel. Then, κ2n = |pn G/p2n G| + κn . Comparing cardinalities shows κ2n = |pn G/p2n G|. But multiplication by pn induces an epimorphism β : G/pn G → pn G/p2n G, g + pn G 7→ pn g + p2n G. Hence κn ≥ |pn G/p2n G| = κ2n > κn , a contradiction. On the other hand, suppose that κn < ∞ for some n > 0. Then, choose n in such a way that κn+1 is infinite. Since n > 0, we have 2n ≥ n + 1, so that κ2n > κn . Arguing as in the last paragraph, we obtain κ2n = |pn G/p2n G| + κn , so pn G/p2n G is infinite. But using the epimorphism β we obtain that pn G/p2n G is finite, a contradiction. Therefore the cardinals κn are all infinite (and equal to κ1 ).  Proposition 4.5. The following are equivalent for an Abelian group A and a prime p: a) A is {Z(p)}-small. b) If B is a bounded p-group, then A is {B}-small. c) A/pA is finite. Proof. The equivalence a) ⇔ c) and the implication b) ⇒ a) are obvious. For c) ⇒ a), note that if A/pA is finite, then by Lemma 4.4, A/pn A is finite for every n. So A is {Z(p)}-small.  Although Z(p∞ ) is clearly not {Z(p∞ )}-small, we have: Proposition 4.6. The following are equivalent for an Abelian group A and a prime p: a) A is {Z(p∞ )}-small. b) If B is a p-group, then A is {B}-small. c) If F is a full free subgroup of A, then (A/F )p is finite. Under these conditions A has finite torsion-free rank. Proof. To see that a) ⇔ b) ⇔ c), it is enough to observe that A is {Z(p∞ )}-small if and only if every torsion epimorphic image of A has the same property, and every kernel corresponding to such image contains a full free subgroup. For the last statement, suppose that A has infinite torsion–free rank and let F be a full free subgroup of A. Then F/pF ≤ A/pF so (A/pF )p is not finite, contradicting c).  Recall that T ∩ R is the class of reduced torsion groups. The symbol P denotes the collection of prime numbers. Corollary 4.7. The following are equivalent for an Abelian group A: a) A is T ∩ R-small. b) A is {Z(p) | p ∈ P}-small;

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c) A = D ⊕ H where D is a divisible group and H is a reduced T ∩ R-small group.  Corollary 4.8. The following are equivalent for a reduced Abelian group A: a) A is T ∩ R-small. b) For every full (free) subgroup F of A, the reduced part of A/F is finite. c) There is a full free subgroup F of A such that the reduced part of A/F is finite.  We obtain an important characterization for self-small torsion groups. Proposition 4.9. [AM75, Proposition 3.1] The following are equivalent for a torsion group T : a) T is small. b) T is T -small. c) T is self-small. d) T is finite.  Furthermore, we obtain the following result generalizing [AM75, Corollary 1.4]: Proposition 4.10. Let A and G be Abelian groups. If Hom(A, G) is countable, then A is {G}-small. Proof. If A is not {G}-small, then there exists a chain U0 ⊂ · · · ⊂ Un ⊂ · · · of proper subgroups of A such that ∪n>0 Un = A and VG (Un ) 6= 0 for all n > 0. Then (VG (Un ))n>0 is a descending chain of subgroups in Hom(A, G). Moreover, we may assume that VG (Un ) 6= VG (Uk ) whenever n 6= k. For every n > 0, we fix a morphism fn ∈ VH (Un ) \ VH (Un+1 ). Observe that, for every x ∈ A, there exists n(x) > 0 such that fn (x) = 0 for all n ≥ n(x). For P every L ⊆ N, we can construct a homomorphism gL : A → G, defined by gL = n∈L fn . Note that gL is actually a finite sum because there exists n(x) > 0 such that fn (x) = 0 for all n ≥ n(x). It is easy to see that for all subsets L 6= K of N, we have gL 6= gK . Thus, Hom(A, G) is not countable.  Lemma 4.11. If a countable Abelian group A is {G}-small then Hom(A, G) is finitely G-cogenerated, i.e., embedded in a finite product of copies of G. Proof. Let A = {a1 , ..., an , ...} be a countable, {G}-small Abelian group. For every n < ω, consider the subgroup An = ha1 , ..., an i. If A is infinite, then (An )n