THE YONEDA EXT AND ARBITRARY COPRODUCTS IN ABELIAN CATEGORIES ALEJANDRO ARGUD´IN MONROY Abstract. There are well known identities involving the Ext bifunctor, co- products, and products in Ab4 abelian categories with enough projectives. Namely, for every such category A, given an object X and a set of ob- n ∼ jects {Ai}i∈I , the following isomorphism can be built ExtA i∈I Ai,X = n n L i∈I ExtA (Ai,X), where Ext is the n-th derived functor of the Hom func- tor.Q The goal of this paper is to show a similar isomorphism for the n-th Yoneda Ext, which is a functor equivalent to Extn that can be defined in more general contexts. The desired isomorphism is constructed explicitly by using colimits, in Ab4 abelian categories with not necessarily enough projec- tives nor injectives, answering a question made by R. Colpi and K R. Fuller in [8]. Furthermore, the isomorphisms constructed are used to characterize Ab4 categories. A dual result is also stated. 1. Introduction The study of extensions is a theory that has developed from multiplicative groups [21, 15], with applications ranging from representations of central simple algebras [4, 13] to topology [10]. In this article we will focus on extensions in an abelian category C. In this context, an extension of an object A by an object C is a short exact sequence 0 → A → M → C → 0 up to equivalence, where two exact sequences are equivalent if there is a morphism arXiv:1904.12182v2 [math.CT] 8 Sep 2020 from one to another with identity morphisms at the ends. This kind of approach was first made by R. Baer in 1934. On his work [1, 2], Baer defined an addition on the class Ext1 (C, A) of extensions of an abelian group A by an abelian group C. His construction can be easily extended to abelian categories, where it is used to 1 show that the class ExtC (C, A) has a natural structure of abelian group. For this 1 reason usually ExtC (C, A) is called the group of extensions of A by C. Later on, H. Cartan and S. Eilenberg [7], using methods of homological algebra, showed that the first derived functor of the Hom C(C, −) functor, or Hom C(−, A) 1 1 functor, is isomorphic to ExtC (C, −), or respectively to ExtC (−, A). This result 1991 Mathematics Subject Classification. Primary 18E99 , 18G15, 18A30. Key words and phrases. Extensions , Yoneda ext , coproducts , products , Ab4 categories. The author thanks the Project PAPIIT-Universidad Nacional Aut´onoma de M´exico IN103317. 1 2 A. ARGUD´IN MONROY marked the beginning of a series of research works looking for ways of constructing the derived functors of the Hom functor without using projective or injective ob- jects, with the spirit that resolutions should be only a calculation tool for derived functors. One of this attempts, registered in the work of D. Buchsbaum, B. Mitchell, S. Schanuel, S. Mac Lane, M.C.R. Butler, and G. Horrocks [16, 5, 6, 17], was based in the ideas of N. Yoneda [23, 24], defining what is known today as the theory of n-extensions and the functor called as the Yoneda Ext. An n-extension of an object A by an object C is an exact sequence of length n 0 → A → M1 →···→ Mn → C → 0 up to equivalence, where the equivalence of exact sequences of length n > 1 is defined in a similar way as was defined for length 1. In this theory, the Baer sum n can be extended to n-extensions, proving that the class ExtC (C, A) of n-extensions of A by C is an abelian group. Recently, the generalization of homological techniques such as Gorenstein or tilting objects to abstract contexts [19, 3, 8, 9], such as abelian categories that do not nec- essarily have projectives or injectives, claim for the introduction of an Ext functor that can be used without restraints. The only problem is that it is not clear if the rich properties of the homological Ext are also valid for the Yoneda Ext. The goal of this work is to make a next step by exploring some properties that the Yoneda Ext shares with the homological Ext. Namely, we will explore the following property that is well known for module cat- egories: Theorem 1.1. [20, Proposition 7.21] Let R be a ring, M ∈ Mod R, and {Ni}i∈I be a set of R-modules. Then, there exist an isomorphism n ∼ n ExtR Ni,M = ExtR (Ni,M) . i∈I ! i∈I M Y The proof of such theorem can be extended to Ab4 abelian categories with enough projectives. Our goal will be to prove an analogue result for the Yoneda Ext without assuming the existence of enough projectives. Let us now describe the contents of this paper. Section 2 is devoted to review the basic results of the theory of extensions by following the steps of B. Mitchell in [17]. In section 3 we prove the desired theorem. More precisely, we show that in an Ab4 abelian category we can build the desired bijections explicitly by using colimits. Finally, in section 4 we use the bijections constructed in section 3 to characterize Ab4 categories. YONEDA EXT 3 2. Extensions In this section we will remember the basic theory of extensions. As was mentioned before, the theory of n-extensions was created by Nobuo Yoneda in [23]. In such paper he worked in a category of modules and most of the results are related with the homological tools built by projective and injective modules. Since our goal is to work in an abelian category without depending on the existence of projective or injective objects, we refer the reader to the work of Barry Mitchell [17] for an approach in abelian categories without further assumptions. Throughout this paper, C will denote an abelian category. Definition 2.1. [17, Section 1] Let C ∈ C, and α : A → B, α′ : A′ → B′ be morphisms in C. We set the following notation: (a) ∇C := ( 1C 1C ): C ⊕ C → C; 1C (b) ∆C := 1C : C → C ⊕ C; ′ α 0 ′ ′ (c) α ⊕ α := ′ : A ⊕ A → B ⊕ B . 0 α 2.1. 1-Extensions. Let us begin by recalling some basic facts and notation about 1-extensions. Definition 2.2. [17, Section 1] Let α : N → N ′, β : M → M ′, and γ : K → K′ be morphisms in C, and consider the following short exact sequences in C f g f ′ g′ η : 0 → N → M → K → 0 and η′ : 0 → N ′ → M ′ → K′ → 0. (a) We say that (α,β,γ): η → η′ is a morphism of short exact sequences if βf = f ′α and γg = g′β. (b) We denote by η ⊕ η′ to the short exact sequence f⊕f ′ g⊕g′ 0 → N ⊕ N ′ → M ⊕ M ′ → K ⊕ K′ → 0. Definition 2.3. [17, Section 1] For N,K ∈ C, let EC(K,N) denote the class of short exact sequences of the form 0 → N → M → K → 0. ′ ′ Remark 2.4. Let A, C ∈C and η, η ∈EC(C, A). Consider the relation η ≡ η given ′ by the existence of a short exact sequence morphism (1A, β, 1C): η → η . By the snake lemma, we know that β is an isomorphism, and hence ≡ is an equivalence relation on EC(C, A). Definition 2.5. [17, Section 1] Consider A, C ∈C. 1 (a) Let ExtC (C, A) := EC(C, A)/ ≡ ; 1 (b) Each object of ExtC (C, A) is refered as an extension from A to C. (c) Every extension from A to C will be denoted with a capital letter E, or by η, in case η is a representative of the class E. 1 ′ 1 ′ ′ (d) Given η ∈ ExtC (C, A) and η ∈ ExtC (C , A ), we will call extension mor- phism from η to η′, to every short exact sequence morphism η → η′. 4 A. ARGUD´IN MONROY (e) If (α,β,γ): E → E′ and (α′,β′,γ′): E′ → E′′ are extension morphisms, we define the composition morphism as (α′,β′,γ′)(α,β,γ) := (α′α, β′β,γ′γ). Remark 2.6. An essential comment made by B. Mitchell in [17] is that the class 1 ExtC (C, A) may not be a set (see [11, Chapter 6, Exercise A] for an example). Considering this fact, we should be cautious when we talk about correspondences between extensions classes. Nevertheless, by simplicity we will say that a corre- spondence 1 ′ ′ 1 Φ : ExtC (C , A ) → ExtC (C, A) 1 ′ ′ is a function, if it associates to each η ∈ ExtC (C , A ) a single element Φ(η) in 1 ExtC (C, A). Remember the following result. Proposition 2.7. [16, Lemma 1.2] Consider a morphism α : X → K and an exact sequence g′ f g 0 → N → M → K → 0 in C. If (E, α′,g′) is 0 N E X 0 α′ α the pullback diagram of the morphisms g and α, g then there is an exact short sequence ηα and a 0 N M K 0 ′ morphism (1, α , α): ηα → η. Of course the construction described above defines a correspondence between the extension classes. ′ Proposition 2.8. [17, Corollary 1.2.] Let η ∈ EC(C, A) and γ ∈ Hom C(C , C). 1 1 ′ Then, the correspondence Φγ : ExtC (C, A) → ExtC (C , A), η 7→ ηγ , is a function. By duality, given a morphism α : N → X and an exact sequence f η : 0 → N → M → K → 0, the pushout of the morphisms f and α, gives us an exact sequence ηα together with a morphism (α, α′, 1) : η → ηα.