The Yoneda Ext and arbitrary coproducts in abelian categories

TL;DR

This paper extends known Ext identities involving coproducts and products to the Yoneda Ext in Ab4 and Ab4* categories without requiring enough projectives or injectives, using limits and colimits explicitly.

Contribution

It establishes explicit isomorphisms for the Yoneda Ext with arbitrary coproducts in broader abelian categories lacking enough projectives or injectives.

Findings

Ext identities hold for Yoneda Ext in more general categories

Explicit constructions of isomorphisms using limits and colimits

Broader applicability in abelian categories without enough projectives/injectives

Abstract

There are well known identities that involve the Ext bifunctor, coproducts, and products in Ab4 and Ab4* abelian categories with enough projectives and enough injectives. Namely, for every such category $\mathcal{A}$, the isomorphisms $\operatorname{Ext}^n (\bigoplus_{i\in I}A_{i},X) \cong \prod_{i\in I} \operatorname{Ext}^n(A_{i},X)$ and $\operatorname{Ext}^n (X,\prod_{i\in I}A_{i}) \cong \prod_{i\in I}\operatorname{Ext}^n (X,A_{i})$ always exist. The goal of this paper is to show similar isomorphisms for the Yoneda Ext in Ab4 and Ab4* abelian categories with not necessarily enough projectives nor injectives. The desired isomorphisms are constructed explicitely by using limits and colimits.

Equations

$$0\to A\to M\to C\to 0$$

$$0\to A\to M_1\to \cdots \to M_n\to C\to 0$$

$${\mathrm{Ext}}_R^n\left(\underset{i\in I}{⨁}{N}_i,M\right)\cong \prod _{i\in I}{\mathrm{Ext}}_R^n\left(N_i,M\right)\text{mbox}.$$

$$\eta :0\to N\stackrel{f}{\to }M\stackrel{g}{\to }K\to 0\text{mbox}and{\eta }^{\prime }:0\to N^{\prime }\stackrel{f^{\prime }}{\to }{M}^{\prime }\stackrel{g^{\prime }}{\to }{K}^{\prime }\to 0\text{mbox}.$$

$$\beta f=f^{\prime }\alpha \text{mbox}and\gamma g=g^{\prime }\beta \text{mbox}.$$

$$0\to N\oplus N^{\prime }\stackrel{f\oplus f^{\prime }}{\to }M\oplus M^{\prime }\stackrel{g\oplus g^{\prime }}{\to }K\oplus K^{\prime }\to 0\text{mbox}.$$

$$({\alpha }^{\prime },{\beta }^{\prime },{\gamma }^{\prime })(\alpha ,\beta ,\gamma ):=({\alpha }^{\prime }\alpha ,{\beta }^{\prime }\beta ,{\gamma }^{\prime }\gamma )\text{mbox}.$$

$$\Phi :{\mathrm{Ext}}_{\mathcal{C}}^1\left(C^{\prime },A^{\prime }\right)\to {\mathrm{Ext}}_{\mathcal{C}}^1\left(C,A\right)$$

$$K$$

$$\eta :0\to N\stackrel{f}{\to }M\to K\to 0\text{mbox},$$

$$\eta :0\to N\stackrel{\mu }{\to }{B}_{n-1}\stackrel{f_{n-1}}{\to }\cdots \stackrel{f_1}{\to }{B}_0\stackrel{\pi }{\to }K\to 0$$

$${\eta }^{\prime }:0\to N^{\prime }\stackrel{{\mu }^{\prime }}{\to }{B}_{n-1}^{\prime }\stackrel{f_{n-1}^{\prime }}{\to }\cdots \stackrel{f_1^{\prime }}{\to }{B}_0^{\prime }\stackrel{{\pi }^{\prime }}{\to }{K}^{\prime }\to 0$$

$${\beta }_{n-1}\mu ={\mu }^{\prime }\alpha \text{mbox},\gamma \pi ={\pi }^{\prime }{\beta }_0\text{mbox}and{\beta }_{i-1}{f}_i=f_i^{\prime }{\beta }_i\forall i\in [0,n-1]\text{mbox}.$$

$$\eta ={\eta }_1\text{mbox},{\eta }_i⪯{\eta }_{i+1}\text{mbox}or{\eta }_{i+1}⪯{\eta }_i,\text{mbox}and{\eta }^{\prime }={\eta }_k\text{mbox}.$$

$$\left(\eta \beta \right){\eta }^{\prime }\to \eta {\eta }^{\prime }\text{mbox}and\eta {\eta }^{\prime }\to \eta \left(\beta {\eta }^{\prime }\right)\text{mbox},$$

$$\eta :0\to N\stackrel{\mu }{\to }{B}_n\to \cdots \to B_1\stackrel{\pi }{\to }K\to 0$$

$${\eta }^{\prime }:0\to K\stackrel{{\mu }^{\prime }}{\to }{B}_m^{\prime }\to \cdots \to B_1^{\prime }\stackrel{{\pi }^{\prime }}{\to }L\to 0$$

$$0\to N\stackrel{\mu }{\to }{B}_n\to \cdots \to B_1\stackrel{{\mu }^{\prime }\pi }{\to }{B}_m^{\prime }\to \cdots \to B_1^{\prime }\stackrel{{\pi }^{\prime }}{\to }L\to 0\text{mbox}.$$

$$\kappa :0\to A\to B_n\to \cdots \to B_1\to C\to 0$$

$${\eta }_i:=0\to K_{i+1}\to B_i\to K_i\to 0\text{mbox},$$

$$\eta :$$

$${\eta }^{\prime }:$$

$$(\alpha ,\beta ,\gamma )=(1,{\beta }^{\prime },\gamma )(\alpha ,{\beta }^{\prime \prime },1)\text{mbox}.$$

$$({\alpha }_i,{\beta }_{i-1},{\alpha }_{i-1}):\overline{\eta {}_i}\to \overline{{\eta }_i^{\prime }}\forall i\in [1,n]$$

$$E+E^{\prime }:={\nabla }_A\left(E\oplus E^{\prime }\right){\Delta }_C\text{mbox}.$$

$$0\to A\stackrel{1}{\to }A\stackrel{0}{\to }\cdots \stackrel{0}{\to }C\stackrel{1}{\to }C\to 0\text{mbox}.$$

$${\left\{{\mu }_i:F(i)\to \mathrm{colim}F\right\}}_{i\in I}\text{mbox},$$

$$F(i)$$

$$u_k:F(k)\to \underset{i\in I}{⨁}F(i)\forall k\in I\text{mbox},v_{\lambda }:F(s(\lambda ))\to \underset{\gamma \in H}{⨁}F(s(\gamma ))\forall \lambda \in H:=\text{mbox}{Hom}_I$$

$$\mathrm{colim}F=\mathrm{Coker}\left(\underset{\gamma \in H}{⨁}F(s(\gamma ))\stackrel{\phi }{\to }\underset{i\in I}{⨁}F(i)\right)\text{mbox},$$

Full text

The author thanks the Project PAPIIT-Universidad Nacional Autónoma de México IN103317.

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 $\mathcal{C}$. In this context, an extension of an object $A$ by an object $C$ is a short exact sequence

[TABLE]

up to equivalence, where two exact sequences are equivalent if there is a morphism 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 $\operatorname{Ext}_{\,}^{1}\left(C,A\right)$ 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 show that the class $\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right)$ has a natural structure of abelian group. For this reason usually $\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right)$ 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 $\operatorname{Hom}{}_{\mathcal{C}}(C,-)$ functor, or $\operatorname{Hom}{}_{\mathcal{C}}(-,A)$ functor, is isomorphic to $\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,-\right)$, or respectively to $\operatorname{Ext}_{\mathcal{C}}^{1}\left(-,A\right)$. This result 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 objects, 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 $n$

[TABLE]

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 can be extended to $n$-extensions, proving that the class $\operatorname{Ext}_{\mathcal{C}}^{n}\left(C,A\right)$ 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 necessarily 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 categories:

Theorem 1.1**.**

[20, Proposition 7.21]** Let $R$ be a ring, $M\in\operatorname{Mod}R$, and $\{N_{i}\}_{i\in I}$ be a set of $R$-modules. Then, there exist an isomorphism

[TABLE]

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.

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, $\mathcal{C}$ will denote an abelian category.

Definition 2.1**.**

[17, Section 1] Let $C\in\mathcal{C}$, and $\alpha:A\rightarrow B$, $\alpha^{\prime}:A^{\prime}\rightarrow B^{\prime}$ be morphisms in $\mathcal{C}$. We set the following notation:

1. (a)

$\nabla_{C}:=\left(\begin{smallmatrix}1_{C}&1_{C}\end{smallmatrix}\right):C\oplus C\rightarrow C\mbox{;}$ 2. (b)

$\Delta_{C}:=\left(\begin{smallmatrix}1_{C}\\ 1_{C}\end{smallmatrix}\right):C\rightarrow C\oplus C\mbox{;}$ 3. (c)

$\alpha\oplus\alpha^{\prime}:=\left(\begin{smallmatrix}\alpha&0\\ 0&\alpha^{\prime}\end{smallmatrix}\right):A\oplus A^{\prime}\rightarrow B\oplus B^{\prime}\mbox{.}$

2.1. 1-Extensions

Let us begin by recalling some basic facts and notation about 1-extensions.

Definition 2.2**.**

[17, Section 1] Let $\alpha:N\rightarrow N^{\prime}$, $\beta:M\rightarrow M^{\prime}$, and $\gamma:K\rightarrow K^{\prime}$ be morphisms in $\mathcal{C}$, and consider the following short exact sequences in $\mathcal{C}$

[TABLE]

1. (a)

We say that $(\alpha,\beta,\gamma):\eta\rightarrow\eta^{\prime}$ is a morphism of short exact sequences if

[TABLE] 2. (b)

We denote by $\eta\oplus\eta^{\prime}$ to the short exact sequence

[TABLE]

Definition 2.3**.**

[17, Section 1] For $N,K\in\mathcal{C}$, let $\mathcal{E}_{\mathcal{C}}(K,N)$ denote the class of short exact sequences of the form $0\rightarrow N\overset{}{\rightarrow}M\overset{}{\rightarrow}K\rightarrow 0\mbox{.}$

Remark 2.4*.*

Let $A,C\in\mathcal{C}$ and $\eta,\eta^{\prime}\in\mathcal{E}_{\mathcal{C}}(C,A)$. Consider the relation $\eta\equiv\eta^{\prime}$ given by the existence of a short exact sequence morphism $(1_{A},\beta,1_{C}):\eta\rightarrow\eta^{\prime}\mbox{.}$ By the snake lemma, we know that $\beta$ is an isomorphism, and hence $\equiv$ is an equivalence relation on $\mathcal{E}_{\mathcal{C}}(C,A)$.

Definition 2.5**.**

[17, Section 1] Consider $A,C\in\mathcal{C}$.

1. (a)

Let $\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right):=\mathcal{E}_{\mathcal{C}}(C,A)/\equiv\mbox{;}$ 2. (b)

Each object of $\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right)$ is refered as an extension from $A$ to $C$. 3. (c)

Every extension from $A$ to $C$ will be denoted with a capital letter $E$, or by $\overline{\eta}$, in case $\eta$ is a representative of the class $E$. 4. (d)

Given $\overline{\eta}\in\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right)$ and $\overline{\eta^{\prime}}\in\operatorname{Ext}_{\mathcal{C}}^{1}\left(C^{\prime},A^{\prime}\right)$, we will call extension morphism from $\overline{\eta}$ to $\overline{\eta^{\prime}}$, to every short exact sequence morphism $\eta\rightarrow\eta^{\prime}$. 5. (e)

If $(\alpha,\beta,\gamma):E\rightarrow E^{\prime}$ and $(\alpha^{\prime},\beta^{\prime},\gamma^{\prime}):E^{\prime}\rightarrow E^{\prime\prime}$ are extension morphisms, we define the composition morphism as

[TABLE]

Remark 2.6*.*

An essential comment made by B. Mitchell in [17] is that the class $\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right)$ 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 correspondence

[TABLE]

is a function, if it associates to each $\overline{\eta}\in\operatorname{Ext}_{\mathcal{C}}^{1}\left(C^{\prime},A^{\prime}\right)$ a single element $\Phi(\overline{\eta})$ in $\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right)$.

Remember the following result.

Proposition 2.7**.**

[16, Lemma 1.2]** Consider a morphism $\alpha:X\rightarrow K$ and an exact sequence $0\rightarrow N\overset{f}{\rightarrow}M\overset{g}{\rightarrow}K\rightarrow 0$ in $\mathcal{C}$. If $(E,\alpha^{\prime},g^{\prime})$ is the pullback diagram of the morphisms $g$ and $\alpha$, then there is an exact short sequence $\eta_{\alpha}$ and a morphism $(1,\alpha^{\prime},\alpha):\eta_{\alpha}\rightarrow\eta$.

K$$M$$N$$X$$E$$N[math][math][math][math]g$$\alpha$$\alpha^{\prime}$$g^{\prime}

Of course the construction described above defines a correspondence between the extension classes.

Proposition 2.8**.**

[17, Corollary 1.2.]** Let $\eta\in\mathcal{E}_{\mathcal{C}}(C,A)$ and $\gamma\in\operatorname{Hom}{}_{\mathcal{C}}(C^{\prime},C)$. Then, the correspondence $\Phi_{\gamma}:\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right)\rightarrow\operatorname{Ext}_{\mathcal{C}}^{1}\left(C^{\prime},A\right)$, $\overline{\eta}\mapsto\overline{\eta_{\gamma}}$, is a function.

By duality, given a morphism $\alpha:N\rightarrow X$ and an exact sequence

[TABLE]

the pushout of the morphisms $f$ and $\alpha$, gives us an exact sequence $\eta^{\alpha}$ together with a morphism $(\alpha,\alpha^{\prime},1):\eta\rightarrow\eta^{\alpha}$. Moreover, we also have that the correspondence $\Phi^{\alpha}:\operatorname{Ext}_{\mathcal{C}}^{1}\left(K,N\right)\rightarrow\operatorname{Ext}_{\mathcal{C}}^{1}\left(K,X\right)$, $\overline{\eta}\mapsto\overline{\eta^{\alpha}}$, is a function.

Definition 2.9**.**

[17, Section 1] For $\alpha:A\rightarrow A^{\prime}$ and $\gamma:C^{\prime}\rightarrow C$ morphisms in $\mathcal{C}$, and $E\in\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right)$, we set $E\gamma:=\Phi_{\gamma}(E)$, and $\alpha E:=\Phi^{\alpha}(E)$.

As we have described, there exists a natural action of the morphisms on the extension classes. These actions are associative and respect identities.

Lemma 2.10**.**

[17, Lemma 1.3]** Let $E\in\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right)$, $\alpha:A\rightarrow A^{\prime}$, $\alpha^{\prime}:A^{\prime}\rightarrow A^{\prime\prime}$, $\gamma:C^{\prime}\rightarrow C$, and $\gamma^{\prime}:C^{\prime\prime}\rightarrow C^{\prime}$ be morphisms in $\mathcal{C}$. Then,

1. (a)

$1_{A}E=E$* and $E1_{C}=E$;* 2. (b)

$(\alpha^{\prime}\alpha)E=\alpha^{\prime}(\alpha E)$* and $E(\gamma\gamma^{\prime})=(E\gamma)\gamma^{\prime}$;* 3. (c)

$(\alpha E)\gamma=\alpha(E\gamma)$.

Next, we recall the definition of the Baer sum.

Definition 2.11**.**

[17, Section 1] For $E,E^{\prime}\in\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right)$, the sum extension of $E$ and $E^{\prime}$ is $E+E^{\prime}:=\nabla_{A}\left(E\oplus E^{\prime}\right)\Delta_{C}\mbox{.}$

This sum operation is well behaved with the actions before described and gives a structure of abelian group to the extension classes.

Theorem 2.12**.**

[17, Lemma 1.4 and Theorem 1.5.]** For any $A,C\in\mathcal{C}$, we have that the pair $\left(\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right),+\right)$ is an abelian group, where the identity element is the extension given by the class of exact sequences that split. Furthermore, let $E\in\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right)$, $E^{\prime}\in\operatorname{Ext}_{\mathcal{C}}^{1}\left(C^{\prime},A^{\prime}\right)$, $\alpha\in\operatorname{Hom}{}_{\mathcal{C}}(A,X)$, $\alpha^{\prime}\in\operatorname{Hom}{}_{\mathcal{C}}(A^{\prime},X^{\prime})$, $\gamma\in\operatorname{Hom}{}_{\mathcal{C}}(Y,C)$ and $\gamma^{\prime}\in\operatorname{Hom}{}_{\mathcal{C}}(Y^{\prime},C^{\prime})$. Then, the following equalities hold true:

1. (a)

$\left(\alpha\oplus\alpha^{\prime}\right)\left(E\oplus E^{\prime}\right)=\alpha E\oplus\alpha^{\prime}E^{\prime}$; 2. (b)

$\left(\alpha+\alpha^{\prime}\right)E=\alpha E+\alpha^{\prime}E$; 3. (c)

$\alpha\left(E+E^{\prime}\right)=\alpha E+\alpha E^{\prime}$; 4. (d)

$\left(E\oplus E^{\prime}\right)\left(\gamma\oplus\gamma^{\prime}\right)=E\gamma\oplus E^{\prime}\gamma^{\prime}$; 5. (e)

$E\left(\gamma+\gamma^{\prime}\right)=E\gamma+E\gamma^{\prime}$; 6. (f)

$\left(E+E^{\prime}\right)\gamma=E\gamma+E^{\prime}\gamma$; 7. (g)

$0E=E0=E_{0}$* for every $E\in\operatorname{Ext}_{\mathcal{C}}^{1}\left(C,A\right)$.*

2.2. $n$-Extensions

We are ready for recalling the definition of $n$-extensions. It is a well known fact that short exact sequences can be sticked together in order to contruct a long exact sequence. Following this thought, the spirit of $n$-extensions is to define a well behaved 1-extensions composition that constructs long extensions.

Definition 2.13**.**

[17, Section 3] We will make use of the following considerations.

1. (a)

For an exact sequence $\eta:\;0\rightarrow A\overset{}{\rightarrow}B_{n-1}\rightarrow\cdots\rightarrow B_{0}\overset{}{\rightarrow}C\rightarrow 0$ in $\mathcal{C}$ we say that $\eta$ is an exact sequence of length $n$, and $A$ and $C$ are the left and right ends of $\eta$, respectively. 2. (b)

Let $\mathcal{E}_{\mathcal{C}}^{n}(L,N)$ denote the class of exact sequences of length $n$ with $L$ and $N$ as right and left ends. 3. (c)

Consider the following exact sequences in $\mathcal{C}$

[TABLE]

A morphism \eta\rightarrow$$\eta^{\prime} is a collection of $n+2$ morphisms $(\alpha,\beta_{n-1},\cdots,\beta_{0},\gamma)$ in $\mathcal{C}$, where $\alpha:N\rightarrow N^{\prime}$, $\gamma:K\rightarrow K^{\prime}$, and $\beta_{i}:B_{i}\rightarrow B^{\prime}_{i}\>\forall i\in[0,n-1]$ are such that

[TABLE]

Equivalently, we can say that a morphism of exact sequences of length $n$ is a commutative diagram

[TABLE]

In the following lines, we define an equivalence relation for studying the classes of exact sequences of length $n$. As we did for the case with $n=1$, we start by saying that two exact sequences $\eta,\eta^{\prime}\in\mathcal{E}_{\mathcal{C}}^{n}(C,A)$ are related, denoted by $\eta\preceq\eta^{\prime}$, if there is a morphism $(1_{A},\beta_{n-1}\cdots,\beta_{0},1_{C}):\eta\rightarrow\eta^{\prime}\mbox{.}$ In this case, we say also that this morphism has fixed ends. Observe that, in contrast with the case $n=1$, this relation needs not to be symmetric. Thus, for achieving our goal, we most consider the equivalence relation $\equiv$ induced by $\preceq$. Namely, we write $\eta\equiv\eta^{\prime}$ if there are exact sequences $\eta_{1},\cdots,\eta_{k}$ such that

[TABLE]

Definition 2.14**.**

[14, Section 9] For $n\geq 1$ and $A,C\in\mathcal{C}$, we consider the class $\operatorname{Ext}_{\mathcal{C}}^{n}\left(C,A\right):=\mathcal{E}_{\mathcal{C}}^{n}(C,A)/\equiv\mbox{,}$ whose elements will be called extensions of length $n$ with $C$ and $A$ as right and left ends. Let $\overline{\eta}$ denote the equivalence class of $\eta\in\mathcal{E}_{\mathcal{C}}^{n}(C,A)$. An extension morphism from $\overline{\eta}$ to $\overline{\eta^{\prime}}$ is just a morphism from $\eta$ to $\eta^{\prime}$.

Remark 2.15*.*

The definition of the equivalence relation above might seem naive. But actually the relation is built with the purpose of making the composition of extensions associate properly when there is a morphism acting in the involved extensions [17, Section 3][16, Section 5]. In the following lines, we will discuss briefly such matter.

Observe how in general, for $\eta\in\mathcal{E}_{\mathcal{C}}^{1}(C,A)$, $\eta^{\prime}\in\mathcal{E}_{\mathcal{C}}^{1}(D,C^{\prime})$ and $\beta:C\rightarrow C^{\prime}$ in $\mathcal{C}$, it is false that $\left(\eta\beta\right)\eta^{\prime}=\eta\left(\beta\eta^{\prime}\right)$. The only affirmation that can be made is that there is an extension morphism $\left(\eta\beta\right)\eta^{\prime}\rightarrow\eta\left(\beta\eta^{\prime}\right)$. To show such morphism, we remember that $\beta$ induces morphisms $\eta\beta\rightarrow\eta\qquad$ and $\qquad\eta^{\prime}\rightarrow\beta\eta^{\prime}\mbox{.}$ Hence, we can build the morphisms

[TABLE]

whose composition gives the wanted morphism. Therefore, even if we have the inequality $\left(\eta\beta\right)\eta^{\prime}\neq\eta\left(\beta\eta^{\prime}\right)$ we can conclude that $\overline{\left(\eta\beta\right)\eta^{\prime}}=\overline{\eta\left(\beta\eta^{\prime}\right)}$.

Definition 2.16**.**

[17, Section 3] Consider the following exact sequences of length $n$ and $m$, respectively

[TABLE]

The composition sequence $\eta\eta^{\prime}$, of $\eta$ with $\eta^{\prime}$, is the exact sequence

[TABLE]

Remark 2.17*.*

Note that each exact sequence in $\mathcal{C}$

[TABLE]

can be written as a composition of $n$ short exact sequences $\kappa=\eta_{n}\cdots\eta_{1}$, where

[TABLE]

with $K_{n+1}:=A$, $K_{1}:=C$ and $K_{i}=\operatorname{Im}(B_{i}\rightarrow B_{i-1})\>\forall i\in[2,n-1]$. We will refer to such factorization of $\kappa$ as its natural decomposition.

Of course, the composition of exact sequences induces a composition of extensions.

Lemma 2.18**.**

[17, Proposition 3.1]** Let $m,n>0$, and $A,C,D\in\mathcal{C}$. Then, the correspondence $\Phi:\operatorname{Ext}_{\mathcal{C}}^{n}\left(C,A\right)\times\operatorname{Ext}_{\mathcal{C}}^{m}\left(D,C\right)\rightarrow\operatorname{Ext}_{\mathcal{C}}^{n+m}\left(D,A\right)$, $(\overline{\eta},\overline{\eta^{\prime}})\mapsto\overline{\eta\eta^{\prime}}$, is a function.

We can now define without ambiguity the composition of extensions.

Definition 2.19**.**

Let $E\in\operatorname{Ext}_{\mathcal{C}}^{n}\left(C,A\right)$ and $E^{\prime}\in\operatorname{Ext}_{\mathcal{C}}^{m}\left(D,C\right)$. For $E=\overline{\eta}$ and $E^{\prime}=\overline{\eta^{\prime}}$, we define the composition extension $EE^{\prime}$ of $E$ with $E^{\prime}$, as the extension $EE^{\prime}:=\overline{\eta\eta^{\prime}}$. If $\eta=\eta_{n}\cdots\eta_{1}$ is the natural decomposition of $\eta$, the induced extension factorization $E=\overline{\eta_{n}}\cdots\overline{\eta_{1}}$ is known as a natural decomposition of $E$.

In the same way, an $n$-extension can be factored into simpler extensions; a morphism of $n$-extensions can be factored into a composition of $n$ simpler morphisms. The next lemma shows the basic fact in this matter.

Lemma 2.20**.**

[17, Lemma 1.1]** Consider a morphism of exact sequences $(\alpha,\beta,\gamma):\eta^{\prime}\rightarrow\eta$, with

$\displaystyle\eta:$ $\displaystyle\quad 0\rightarrow A\overset{f}{\rightarrow}B\overset{g}{\rightarrow}C\rightarrow 0\mbox{ and }$

$\displaystyle\eta^{\prime}:$ $\displaystyle\quad 0\rightarrow A^{\prime}\overset{f^{\prime}}{\rightarrow}B^{\prime}\overset{g^{\prime}}{\rightarrow}C^{\prime}\rightarrow 0\mbox{.}$

Then, $\eta\gamma=\alpha\eta^{\prime}$ and $(\alpha,\beta,\gamma)$ factors through $\eta\gamma$ as

$(\alpha,\beta,\gamma)=(1,\beta^{\prime},\gamma)(\alpha,\beta^{\prime\prime},1)\mbox{.}$

[math]A^{\prime}$$B^{\prime}$$C^{\prime}[math][math]A$$E$$C^{\prime}[math][math]A$$B$$C[math]f^{\prime}$$g^{\prime}$$f$$g$$\alpha$$\beta^{\prime\prime}$$\beta^{\prime}$$\gamma

In general, we can make the following affirmation.

Corollary 2.21**.**

Let $\eta,\eta^{\prime}\in\mathcal{E}_{\mathcal{C}}^{n}(C,A)$ be exact sequences with natural decompositions $\eta=\eta_{n}\cdots\eta_{1}$ and $\eta^{\prime}=\eta^{\prime}_{n}\cdots\eta^{\prime}_{1}$. Then, the following statements hold true.

1. (a)

There is an exact sequence morphism $(\alpha,\beta_{n-1},\cdots,\beta_{0},\gamma):\eta\rightarrow\eta^{\prime}$ if, and only if, there is a collection of extension morphisms

[TABLE]

where $\alpha_{n}=\alpha$ and $\alpha_{0}=\gamma$. 2. (b)

If there is an exact sequence morphism $(\alpha,\beta_{n-1},\cdots,\beta_{0},\gamma):\eta\rightarrow\eta^{\prime}$, then there is a collection of morphisms $\alpha_{n-1},\cdots,\alpha_{1}$ in $\mathcal{C}$ satisfying the following equalities:

1. (b1)

$\overline{\eta^{\prime}_{n}}\cdots\overline{\eta^{\prime}_{1}}\gamma=\alpha\overline{\eta_{n}}\cdots\overline{\eta_{1}}$, 2. (b2)

$\overline{\eta^{\prime}_{i}}\cdots\overline{\eta^{\prime}_{1}}\gamma=\alpha_{i}\overline{\eta_{i}}\cdots\overline{\eta_{1}}\>\forall i\in[1,n-1]$, and 3. (b3)

$\overline{\eta^{\prime}_{n}}\cdots\overline{\eta^{\prime}_{i+1}}\alpha_{i}=\alpha\overline{\eta_{n}}\cdots\overline{\eta_{i+1}}\>\forall i\in[1,n-1]$.

Proof.

It follows from 2.20. ∎

By Lemma 2.18, the following actions are well defined.

Definition 2.22**.**

[17, Section 3] Consider $\eta,\eta^{\prime}\in\mathcal{E}_{\mathcal{C}}^{n}(C,A)$, $E:=\overline{\eta}\in\operatorname{Ext}_{\mathcal{C}}^{n}\left(C,A\right)$, $E^{\prime}:=\overline{\eta^{\prime}}\in\operatorname{Ext}_{\mathcal{C}}^{n}\left(C,A\right)$, and let $\eta=\eta_{n}\cdots\eta_{1}$ and $\eta^{\prime}=\eta^{\prime}_{n}\cdots\eta^{\prime}_{1}$ be the natural decompositions of $\eta$ and $\eta^{\prime}$.

1. (a)

Given $\alpha\in\operatorname{Hom}{}_{\mathcal{C}}(A,A^{\prime})$, we define $\alpha E:=\alpha\overline{\eta_{n}}\cdots\overline{\eta_{1}}\mbox{.}$ 2. (b)

Given $\gamma\in\operatorname{Hom}{}_{\mathcal{C}}(C^{\prime},C)$, we define $E\gamma:=\overline{\eta_{n}}\cdots\overline{\eta_{1}}\gamma\mbox{.}$ 3. (c)

We define the sum of extensions of length $n$ in the following way

[TABLE]

Most of the properties, proved earlier for extensions of length 1, can be naturally extended, as can be seen in the following lines.

Corollary 2.23**.**

[17, Lemma 3.2 an Theorem 3.3]** Let $n>0$.

1. (a)

Let $E\in\operatorname{Ext}_{\mathcal{C}}^{n}\left(C,A\right)$, $E^{\prime}\in\operatorname{Ext}_{\mathcal{C}}^{m}\left(D,C^{\prime}\right)$, $\beta\in\operatorname{Hom}{}_{\mathcal{C}}(C^{\prime},C)$, $\beta^{\prime}\in\operatorname{Hom}{}_{\mathcal{C}}(C^{\prime\prime},C^{\prime})$, $\alpha\in\operatorname{Hom}{}_{\mathcal{C}}(A,A^{\prime})$, and $\alpha^{\prime}\in\operatorname{Hom}{}_{\mathcal{C}}(A^{\prime},A^{\prime\prime})$. Then the following equalities hold true:

1. (a1)

$\left(E\beta\right)E^{\prime}=E\left(\beta E^{\prime}\right)$; 2. (a2)

$1_{A}E=E=E1_{C}$; 3. (a3)

$E\left(\beta\beta^{\prime}\right)=\left(E\beta\right)\beta^{\prime}$; 4. (a4)

$\left(\alpha^{\prime}\alpha\right)E=\alpha^{\prime}\left(\alpha E\right)$. 2. (b)

Let $E\in\operatorname{Ext}_{\mathcal{C}}^{n}\left(C,A\right)$, $E^{\prime}\in\operatorname{Ext}_{\mathcal{C}}^{n}\left(C^{\prime},A^{\prime}\right)$, $F\in\operatorname{Ext}_{\mathcal{C}}^{m}\left(D,C\right)$, $F^{\prime}\in\operatorname{Ext}_{\mathcal{C}}^{m}\left(D^{\prime},C^{\prime}\right)$, $\alpha\in\operatorname{Hom}{}_{\mathcal{C}}(A,X)$, $\alpha^{\prime}\in\operatorname{Hom}{}_{\mathcal{C}}(A^{\prime},X^{\prime})$, $\gamma\in\operatorname{Hom}{}_{\mathcal{C}}(Y,C)$, and $\gamma^{\prime}\in\operatorname{Hom}{}_{\mathcal{C}}(Y^{\prime},C^{\prime})$. Then the following equalities hold true:

1. (b1)

$(\alpha\oplus\alpha^{\prime})\left(E\oplus E^{\prime}\right)=\alpha E\oplus\alpha^{\prime}E^{\prime}$* and $\left(E\oplus E^{\prime}\right)\left(\gamma\oplus\gamma^{\prime}\right)=E\gamma\oplus E^{\prime}\gamma^{\prime}$;* 2. (b2)

$\left(E\oplus E^{\prime}\right)\left(F\oplus F^{\prime}\right)=EF\oplus E^{\prime}F^{\prime}$; 3. (b3)

$\left(E+E^{\prime}\right)F=EF+E^{\prime}F$* and $E\left(F+F^{\prime}\right)=EF+EF^{\prime}$;* 4. (b4)

$(\alpha+\alpha^{\prime})E=\alpha E+\alpha^{\prime}E$* and $E\left(\gamma+\gamma^{\prime}\right)=E\gamma+E\gamma^{\prime}$; and* 5. (b5)

$\alpha\left(E+E^{\prime}\right)=\alpha E+\alpha E^{\prime}$* and $\left(E+E^{\prime}\right)\gamma=E\gamma+E^{\prime}\gamma$.* 3. (c)

The pair $\left(\operatorname{Ext}_{\mathcal{C}}^{n}\left(C,A\right),+\right)$ is an abelian group, where the identity element is the extension $E_{0}$ given by the exact sequence, in case $n\geq 2$,

[TABLE]

We conclude this section with the following theorem that focus on characterizing the trivial extensions.

Theorem 2.24**.**

[17, Theorem 4.2]** Let $n>1$ and $\eta\in\mathcal{E}_{\mathcal{C}}^{n}(C,A)$ with a natural decomposition $\eta=\eta_{n}\cdots\eta_{1}$. Then , the following statements hold true:

1. (a)

$\overline{\eta}=0$; 2. (b)

there is an exact sequence \kappa$$\in\mathcal{E}_{\mathcal{C}}^{n}(C,A) and a pair of morphisms with fixed ends $0\leftarrow\kappa\rightarrow\eta\mbox{.}$ 3. (c)

there is an exact sequence $\kappa^{\prime}\in\mathcal{E}_{\mathcal{C}}^{n}(C,A)$ and a pair of morphisms with fixed ends $0\rightarrow\kappa^{\prime}\leftarrow\eta\mbox{.}$

3. Additional structure in Abelian Categories

In this section we will approach our problem dealing with arbitrary products and coproducts. Of course, an abelian category does not necessarily have arbitrary products and coproducts. Hence, we will review briefly the theory of abelian categories with additional structure introduced by A. Grothendieck in [12]. For further reading we suggest [18, Section 2.8].

3.1. Limits and colimits

Definition 3.1**.**

[18, Section 1.4.] Let $\mathcal{C}$ and $I$ be categories, where $I$ is small (that is the class of objects of $I$ is a set). Let $F:I\rightarrow\mathcal{C}$ be a functor and $X\in\mathcal{C}$. A family of morphisms $\left\{\alpha_{i}:F(i)\rightarrow X\right\}_{i\in I}$ in $\mathcal{C}$ is co-compatible with $F$, if $\alpha_{i}=\alpha_{j}F(\lambda)$ for every $\lambda:i\rightarrow j$ in $I$.

The colimit (or inductive limit) of $F$ is an object $\operatorname{colim}F$ in $\mathcal{C}$ with a co-compatible family of morphisms

$\left\{\mu_{i}:F(i)\rightarrow\operatorname{colim}F\right\}_{i\in I}\mbox{,}$

such that for every co-compatible family of morphisms $\left\{\gamma_{i}:F(i)\rightarrow X\right\}_{i\in I}$, there is a unique morphism $\gamma:\operatorname{colim}F\rightarrow X$ such that $\gamma_{i}=\gamma\mu_{i}$ for every $i\in I$.

F(i)$$F(j)$$X$$\operatorname{colim}F$$F(\lambda)$$\gamma_{i}$$\gamma_{j}$$\mu_{i}$$\mu_{j}$$\gamma

Let $I$ be a small category and $\lambda:i\rightarrow j$ be a morphism in $I$. The following notation will be useful $s(\lambda):=i$ and $t(\lambda):=j$.

Proposition 3.2**.**

[22, Proposition 8.4]** Let $\mathcal{C}$ be a preadditive category with coproducts and cokernels, $I$ be a small category, $F:I\rightarrow\mathcal{C}$ be a functor, and

[TABLE]

*be the respective canonical inclusions into the coproducts. Then,

$\operatorname{colim}F=\operatorname{Coker}\left(\bigoplus_{\gamma\in H}F(s(\gamma))\overset{\varphi}{\rightarrow}\bigoplus_{i\in I}F(i)\right)\mbox{,}$

where $\varphi$ is the morphism induced by the universal property of coproducts applied to the family of morphisms

$\left\{\varphi_{\lambda}:=u_{s(\lambda)}-u_{t(\lambda)}F(\lambda)\right\}_{\lambda\in H}\mbox{.}$

\bigoplus_{\gamma\in H}F(s(\gamma))$$\bigoplus_{i\in I}F(i)$$F(s(\lambda))$$F(s(\lambda))\oplus F(t(\lambda))$$\varphi$$\left(\begin{smallmatrix}1\\ -F(\lambda)\end{smallmatrix}\right)$$v_{\lambda}$$\left(\begin{smallmatrix}u_{s(\lambda)}&u_{t(\lambda)}\end{smallmatrix}\right)

The dual notion of colimit is the limit.

Definition 3.3**.**

[18, Section 1.4.] Let $\mathcal{C}$ and $I$ be categories, with $I$ small. Let $F:I\rightarrow\mathcal{C}$ be a functor and $X\in\mathcal{C}$. A family of morphisms $\left\{\alpha_{i}:X\rightarrow F(i)\right\}_{i\in I}$ in $\mathcal{C}$ is compatible with $F$, if $\alpha_{j}=F(\lambda)\alpha_{i}$ for every $\lambda:i\rightarrow j$ in $I$.

The limit (or projective limit) of $F$ is an object $\lim F$ in $\mathcal{C}$ together with a compatible family of morphisms

$\left\{\mu_{i}:\lim F\rightarrow F(i)\right\}_{i\in I}$

such that for any compatible family of morphisms $\left\{\gamma_{i}:X\rightarrow F(i)\right\}_{i\in I}$ there is a unique $\gamma\in\operatorname{Hom}_{\mathcal{C}}(X,\lim F)$ such that $\gamma_{i}=\mu_{i}\gamma$ for every $i\in I$.

F(i)$$F(j)$$X$$\operatorname{lim}F$$F(\lambda)$$\gamma_{i}$$\gamma_{j}$$\mu_{i}$$\mu_{j}$$\gamma

Proposition 3.4**.**

[22, Proposition 8.2]** Let $\mathcal{C}$ be a preadditive category with products and kernels, $I$ be an small category, $F:I\rightarrow\mathcal{C}$ be a functor, and

[TABLE]

*be the respective canonical proyections out of the products. Then,

$\lim F=\operatorname{Ker}\left(\prod_{i\in I}F(i)\overset{\varphi}{\rightarrow}\prod_{\gamma\in H}F(t(\gamma))\right)\mbox{,}$

where $\varphi$ is the morphism induced by the universal property of products applied to the family of morphisms

$\left\{\varphi_{\lambda}:=F(\lambda)u_{s(\lambda)}-u_{t(\lambda)}\right\}_{\lambda\in H}\mbox{.}$

\prod_{i\in I}F(i)$$\prod_{\gamma\in H}F(t(\gamma))$$F(s(\lambda))\oplus F(t(\lambda))$$F(t(\lambda))$$\varphi$$\left(\begin{smallmatrix}u_{s(\lambda)}\\ u_{t(\lambda)}\end{smallmatrix}\right)$$v_{\lambda}$$\left(\begin{smallmatrix}-F(\lambda)&1\end{smallmatrix}\right)

Definition 3.5**.**

Let $I$ be a small category and $\mathcal{C}$ be an abelian category. It is said that a family of objects and morphisms $\left(M_{i},f_{\alpha}\right)_{i\in I,\alpha\in\mbox{Hom}_{I}}$ is a direct system, if there is a functor $F:I\rightarrow\mathcal{C}$ such that $F(i)=M_{i}\,\forall i\in I$ and $F(\alpha)=f_{\alpha}$ for every $\alpha\in\mbox{Hom}_{I}$.

3.2. Ab3 and Ab4 Categories

Definition 3.6**.**

[18, Section 2.8.] An Ab3 category is an abelian category satisfying the following condition:

**(Ab3): **

For every set of objects $\left\{A_{i}\right\}_{i\in I}$ in $\mathcal{C}$, the coproduct $\bigoplus_{i\in I}A_{i}$ exists.

We remember the following well known fact.

Proposition 3.7**.**

[18, Section 2.8.]** Let $\mathcal{C}$ be an Ab3 category and

[TABLE]

be a set of exact sequences in $\mathcal{C}$. Then,

[TABLE]

is an exact sequence in $\mathcal{C}$.

In general, it is not possible to prove that $\bigoplus_{i\in I}f_{i}$ is a monomorphism if each $f_{i}$ is a monomorphism. For this reason, the following Grothendieck’s condition arised.

Definition 3.8**.**

[18, Proposition 8.3.] An Ab4 category is an Ab3 category $\mathcal{C}$ satisfying the following condition:

**(Ab4): **

for every set of monomorphisms $\left\{f_{i}:X_{i}\rightarrow Y_{i}\right\}_{i\in I}$ in $\mathcal{C}$, the morphism $\bigoplus_{i\in I}f_{i}$ is a monomorphism.

We will refer to the dual condition as Ab4*.

Remark 3.9*.*

Let $\mathcal{C}$ be an Ab4 category. Then, for every sets of objects $\{A_{i}\}_{i\in I}$ and $\{B_{i}\}_{i\in I}$ in $\mathcal{C}$, the correspondence

[TABLE]

is a well defined morphism of abelian groups.

3.3. Ext groups and arbitrary products and coproducts

We are finally ready to proceed in our goal’s direction.

Lemma 3.10**.**

Let $\mathcal{C}$ be an Ab4 category, and

[TABLE]

be a set of short exact sequences in $\mathcal{C}$. Then, there is a short exact sequence

[TABLE]

such that $\eta\mu_{i}=\eta_{i}\>\forall i\in I$, where $\left\{\mu_{i}:C_{i}\rightarrow\bigoplus_{i\in I}C_{i}\right\}_{i\in I}$ is the family of canonical inyections into the coproduct.

Proof.

Consider the set $\left\{f_{i}:B\rightarrow A_{i}\right\}_{i\in I}$ as a direct system. Observe that the set of morphisms of exact sequences

[TABLE]

is a direct system of exact sequences. We will consider the colimit of such system and prove that, as result, we get a short exact sequence. To this end, we observe that $(B,1_{i}:B\rightarrow B)_{i\in I}$ is the colimit of the system $\left\{1_{i}:B\rightarrow B\right\}_{i\in I}$ and that $(\bigoplus_{i\in I}C_{i},\mu_{i}:C_{i}\rightarrow\bigoplus_{i\in I}C_{i})$ is the colimit of the system $\left\{0:0\rightarrow C_{i}\right\}_{i\in I}$.

Hence, by 3.2 we build the diagram beside, where the columns are the morphism mentioned in 3.2, the upper and central rows are coproducts of the sequences $\beta$ and $\eta_{i}$ respectively, and the bottom row is the result of the colimits. Thus, by the snake lemma we get the exact sequence

\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle\bigoplus_{i\in I}B_{i}$$\scriptstyle\bigoplus_{i\in I}B_{i}$$\scriptstyle 0$$\scriptstyle B\oplus(\bigoplus_{i\in I}B_{i})$$\scriptstyle B\oplus(\bigoplus_{i\in I}A_{i})$$\scriptstyle\bigoplus_{i\in I}C_{i}$$\scriptstyle B$$\scriptstyle\operatorname{colim}(f_{i})$$\scriptstyle\bigoplus_{i\in I}C_{i}

[TABLE]

Furthermore, the families of morphisms associated to such colimits give us the exact sequence morphisms $(1,\mu^{\prime}_{i},\mu_{i}):\eta_{i}\rightarrow\eta\;\forall i\in I\mbox{,}$ which proves the statement. ∎

Proposition 3.11**.**

Let $\mathcal{C}$ be an Ab4 category and $\{A_{i}\}_{i\in I}$ a set of objects in $\mathcal{C}$. Consider the coproduct with the canonical inclusions $\left(\mu_{i}:A_{i}\rightarrow\bigoplus_{i\in I}A_{i}\right)_{i\in I}$. Then, the correspondence $\Psi:\operatorname{Ext}_{\mathcal{C}}^{1}\left(\bigoplus A_{i},B\right)\rightarrow\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{1}\left(A_{i},B\right)$, defined by $E\mapsto\left(E\mu_{i}\right)_{i\in I}$, is an isomorphism for every $B\in\mathcal{C}$.

Proof.

We will proceed by proving the following steps:

1. (a)

The correspondence $\Psi$ is an abelian group morphism. 2. (b)

$\Psi$ is injective. 3. (c)

Given $(\overline{\eta_{i}})\in\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{1}\left(A_{i},B\right)$, there is $E\in\operatorname{Ext}_{\mathcal{C}}^{1}\left(\bigoplus A_{i},B\right)$ such that $\Psi(E)=(\overline{\eta_{i}})$.

Clearly, proving these statements are enough to conclude the desired proposition.

1. (a)

It follows by 2.21. 2. (b)

Suppose that $E$ is an extension with representative

[TABLE]

such that $E\mu_{i}=0$ $\forall i\in I$. Suppose that $(1,p_{i},\mu_{i}):E\mu_{i}\rightarrow E$ is the morphism induced by $\mu_{i}$, and that each extension $E\mu_{i}$ has as representative the exact

sequence $\eta_{i}:\;0\rightarrow B\overset{f_{i}}{\rightarrow}C_{i}\overset{g_{i}}{\rightarrow}A_{i}\rightarrow 0\mbox{.}$ By definition, there is a morphism $h_{i}:A_{i}\rightarrow C_{i}$ such that $g_{i}h_{i}=1_{A_{i}}$. Thus, by the coproduct universal property, there is a unique morphism $h:\bigoplus_{i\in I}A_{i}\rightarrow C$ such that $h\mu_{i}=p_{i}h_{i}$ $\forall i\in\{1,2\}$. Therefore, by

$gh\mu_{i}=gp_{i}h_{i}=\mu_{i}g_{i}h_{i}=\mu_{i}\>\forall i\in I,$

[math]B$$C_{i}$$A_{i}[math][math]B$$C$$\bigoplus_{i\in I}A_{i}[math]f_{i}$$g_{i}$$f$$g$$p_{i}$$\mu_{i}$$h_{i}$$h

we have that $gh=1_{\bigoplus_{i\in I}A_{i}}$ by the coproduct universal property; and thus, $E=0$. 3. (c)

It follows by 3.10.

∎

Theorem 3.12**.**

Let $\mathcal{C}$ be an Ab4 category, $n\geq 1$, and $\{A_{i}\}_{i\in I}$ be a set of objects in $\mathcal{C}$. Consider the coproduct $\bigoplus_{i\in I}A_{i}$ and the canonic inclusions $\left(\mu_{i}:A_{i}\rightarrow\bigoplus_{i\in I}A_{i}\right)_{i\in I}$. Then, the correspondence $\Psi_{n}:\operatorname{Ext}_{\mathcal{C}}^{n}\left(\bigoplus A_{i},B\right)\rightarrow\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{n}\left(A_{i},B\right)$, $E\mapsto\left(E\mu_{i}\right)_{i\in I}$, is an isomorphism of abelian groups for every $B\in\mathcal{C}$.

Proof.

We will proceed by proving the following statements:

1. (a)

The correspondence $\Psi_{n}$ is a morphism of abelian groups; 2. (b)

$\Psi_{n}$ is injective; 3. (c)

For every $(\overline{\eta_{i}})\in\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{n}\left(A_{i},B\right)$, there is $E\in\operatorname{Ext}_{\mathcal{C}}^{n}\left(\bigoplus_{i\in I}A_{i},B\right)$ such that $\Psi_{n}(E)=(\overline{\eta_{i}})$.

It is worth to mention that the result was already proved in 3.11 for $n=1$. Furthermore, in the proof of 3.11(c) it was shown explicitly the inverse function of $\Psi_{1}$. We will denote such correspondence as $\Psi_{1}^{-1}$.

1. (a)

It follows by 2.21. 2. (b)

Let $\overline{\eta}$ be an extension with a natural decomposition $\overline{\eta}=\overline{\eta_{n}}\cdots\overline{\eta_{1}}$ such that $\overline{\eta}\mu_{i}=0\>\forall i\in I$. By 2.24 this means that for every $i\in I$ there is a pair of exact sequences morphisms with fixed ends $\eta\mu_{i}\leftarrow\kappa_{i}\rightarrow 0$. Suppose that each exact sequence $\kappa_{i}$ has the natural decomposition $\kappa_{i}=\kappa(i)_{n}\cdots\kappa(i)_{1}\mbox{.}$ It follows from the morphism $\kappa_{i}\rightarrow 0$ that

[TABLE]

is a splitting exact sequence. Let $(\overline{\kappa^{\prime}_{i}}):=(\overline{\kappa^{\prime}_{i}})_{i\in I}\in\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{1}\left(X_{i},B\right)\mbox{.}$ By 3.11(c), we know that $\Psi_{1}^{-1}(\overline{\kappa^{\prime}_{i}})\in\operatorname{Ext}_{\mathcal{C}}^{1}\left(\bigoplus_{i\in I}X_{i},B\right)$ is an extension such that $\Psi_{1}^{-1}(\overline{\kappa^{\prime}_{i}})\mu^{\prime}_{i}=\kappa^{\prime}_{i}\forall i\in I$, where each $\mu^{\prime}_{i}:X_{i}\rightarrow\bigoplus_{i\in I}X_{i}$ is the canonical inclusion. Let $\kappa:=\Psi_{1}^{-1}(\overline{\kappa^{\prime}_{i}})\left(\overline{\bigoplus_{i\in I}\kappa(i)_{n-1}}\right)\cdots\left(\overline{\bigoplus_{i\in I}\kappa(i)_{1}}\right)\mbox{.}$ We will show that there is a pair of exact sequence morphisms with fixed ends $\eta\leftarrow\kappa\rightarrow 0$, which will prove (b) by 2.24. Indeed, by the fact that for every $i\in I$ there is a morphism with fixed ends $\eta\mu_{i}\leftarrow\kappa_{i}$, it follows that there is a morphism with fixed right end $\eta_{n-1}\cdots\eta_{1}\mu_{i}\leftarrow\kappa(i)_{n-1}\cdots\kappa(i)_{1}\mbox{,}$ inducing by the coproduct universal property a morphism with fixed right end

[TABLE]

Furthermore, by the proof of 3.11 we know that $\Psi_{1}^{-1}(\overline{\kappa^{\prime}_{i}})$ has as representative the exact sequence $0\rightarrow B\overset{f}{\rightarrow}\mbox{colim}(f_{i})\overset{g}{\rightarrow}\bigoplus_{i\in I}X_{i}\rightarrow 0\mbox{.}$ Hence, using the colimit universal property, is easy to see that there is a morphism with fixed left end $\eta_{n}\leftarrow\Psi_{1}^{-1}(\overline{\kappa^{\prime}_{i}})\mbox{.}$ Therefore, with the last morphisms we can build a morphism with fixed ends $\eta\leftarrow\kappa\mbox{.}$ For showing the existence of a morphism with fixed ends $\kappa\rightarrow 0$, it is enough to show that $f$ is a splitting monomorphism, which follows straightforward from the colimit universal property together with the fact that every $f_{i}$ is a splitting monomorphism. 3. (c)

Let $(\overline{\eta_{i}})\in\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{n}\left(A_{i},B\right)$. We observe the following facts for every $i\in I$. Suppose $\overline{\eta_{i}}=\overline{\kappa_{n}^{i}}\cdots\overline{\kappa_{1}^{i}}$ is a natural decomposition, where

[TABLE]

Consider the coproduct canonical inclusions $u_{k}^{i}:B_{k}^{i}\rightarrow\bigoplus_{i\in I}B_{k}^{i}$. Observe that $u_{1}^{i}=\mu_{i}\,\forall i\in I$. By 2.20 we can see that $\overline{\left(\bigoplus_{i\in I}\kappa_{k}^{i}\right)}u_{k}^{i}=u_{k+1}^{i}\overline{\kappa_{k}^{i}}$ for all $k\in\left\{1,\cdots,n+1\right\}\mbox{.}$ Hence, by 3.11(c), for every $i\in I$ the extension defined as $\overline{\eta}:=\Psi_{1}^{-1}(\kappa_{n}^{i})_{i\in I}\overline{\left(\bigoplus_{i\in I}\kappa^{i}{}_{n-1}\right)}\cdots\overline{\left(\bigoplus_{i\in I}\kappa^{i}{}_{1}\right)}\mbox{,}$ satisfies by recursion the following equalities

[TABLE]

∎

By duality we have the following result.

Theorem 3.13**.**

Let $\mathcal{C}$ be an Ab4 category, $n\geq 1$, and $\{A_{i}\}_{i\in I}$ be a set of objects in $\mathcal{C}$. Consider the product $\left(\pi_{i}:\prod_{i\in I}A_{i}\rightarrow A_{i}\right)_{i\in I}$. Then, the correspondence $\Phi_{n}:\operatorname{Ext}_{\mathcal{C}}^{n}\left(B,\prod_{i\in I}A_{i}\right)\rightarrow\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{n}\left(B,A_{i}\right)$, $E\mapsto\left(\pi_{i}E\right)_{i\in I}$, is an isomorphism of abelian groups for every $B\in\mathcal{C}$.*

We will end this section introducing an application related to the tilting theory developed in recent years. Namely, R. Colpi and K. R. Fuller developed a theory of tilting objects of projective dimension $\leq 1$ for abelian categories in [8], and P. Čoupek and J. Št’ovíček developed a theory of cotilting objects of injective dimension $\leq 1$ for Grothendieck categories in [9]. A fundamental result needed in these theories is that

[TABLE]

Such result is proved showing that, in any Ab3 abelian category $\mathcal{A}$, there is an injective correspondence $\mbox{Ext}_{\mathcal{A}}^{1}\left(\bigoplus_{i\in I}A_{i},X\right)\rightarrow\prod_{i\in I}\mbox{Ext}_{\mathcal{A}}^{1}(A_{i},X)$ (see [8, Proposition 8.1, Proposition 8.2] and [9, Proposition A.1] or the proof of 3.11). Now, for extending the theory to tilting objects of projective dimension $\leq n$, it is needed a similar result for $\mbox{Ext}^{n}$. But, it is not known in general if there is an injective correspondence $\mbox{Ext}_{\mathcal{A}}^{n}\left(\bigoplus_{i\in I}A_{i},X\right)\rightarrow\prod_{i\in I}\mbox{Ext}_{\mathcal{A}}^{n}(A_{i},X)$.

The following result follows from 3.13 and 3.12. It is worth to mention that it extends [8, Corollary 8.3] and the dual of [9, Corollary A.2] when the category is Ab4.

Corollary 3.14**.**

Let $\mathcal{C}$ be an abelian category, $n\geq 1$, $\{A_{i}\}_{i\in I}$ be a set of objects in $\mathcal{C}$, and $B\in\mathcal{C}$. Then, the following statements hold true:

1. (a)

If $\mathcal{C}$ is Ab4, then $\operatorname{Ext}_{\mathcal{C}}^{n}\left(\bigoplus_{i\in I}A_{i},B\right)=0$ if and only if $\operatorname{Ext}_{\mathcal{C}}^{n}\left(A_{i},B\right)=0\,\forall i\in I$. 2. (b)

If $\mathcal{C}$ is Ab4, then $\operatorname{Ext}_{\mathcal{C}}^{n}\left(B,\prod_{i\in I}A_{i}\right)=0$ if and only if $\operatorname{Ext}_{\mathcal{C}}^{n}\left(B,A_{i}\right)=0\,\forall i\in I$.*

4. A characterization of Ab4

This section is inspired by the comments made by Sergio Estrada during the Coloquio Latinoamericano de Álgebra XXIII. The goal is to prove that if the correspondence $\Psi:\operatorname{Ext}_{\mathcal{C}}^{1}\left(\bigoplus A_{i},B\right)\rightarrow\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{1}\left(A_{i},B\right)$ defined above is always biyective for an Ab3 category $\mathcal{C}$, then $\mathcal{C}$ is Ab4.

Throughout this section for every natural number $n>0$ we will consider the correspondence $\Psi_{n}:\operatorname{Ext}_{\mathcal{C}}^{n}\left(\bigoplus X_{i},Y\right)\rightarrow\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{n}\left(X_{i},Y\right)$ defined above.

In 3.10, it was proved that, if $\mathcal{C}$ is Ab4, then given a set of exact sequences

[TABLE]

it can be built an exact sequence $0\rightarrow B\overset{f}{\rightarrow}\mbox{colim}(f_{i})\overset{}{\rightarrow}\bigoplus_{i\in I}C_{i}\rightarrow 0$, where $f$ is part of the co-compatible family of morphisms associated to $\operatorname{colim}f_{i}$. In case $\mathcal{C}$ is only an Ab3 category, then by doing a similar construction we get an exact sequence $B\rightarrow\operatorname{colim}f_{i}\rightarrow\bigoplus_{i\in I}C_{i}\rightarrow 0$. Indeed, consider the direct system of exact sequences

[math]B$$B[math][math][math]B$$A_{i}$$C_{i}[math]1$$f_{i}$$1$$f_{i}

Then, we have an exact sequence $B\stackrel{{\scriptstyle f}}{{\rightarrow}}\operatorname{colim}f_{i}\stackrel{{\scriptstyle g}}{{\rightarrow}}\bigoplus_{i\in I}C_{i}\rightarrow 0$, where $f$ and $g$ are induced by the colimit universal property (see [18, page 55]). Such exact sequence we shall name it $\Theta(\eta_{i})$.

As a first step we will show that, even if the category is not Ab4, if the correspondence $\Psi$ is biyective, then the inverse correspondence is given by $\Theta$. That is, if $\Psi:\operatorname{Ext}_{\mathcal{C}}^{1}\left(\bigoplus A_{i},B\right)\rightarrow\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{1}\left(A_{i},B\right)$ is biyective, then for every set of exact sequences $\left\{\eta_{i}:\>0\rightarrow B\overset{f_{i}}{\rightarrow}A_{i}\overset{}{\rightarrow}C_{i}\rightarrow 0\right\}_{i\in I}$, then the morphism $f$ in $\Phi(\eta_{i})$ is monic, and $\Psi\overline{\Theta(\eta_{i})}=1$.

Lemma 4.1**.**

Let $\mathcal{C}$ be an AB3 category, $\left\{A_{i}\right\}_{i\in I}$ be a set of objects in $\mathcal{C}$, and $B\in\mathcal{C}$. Consider the coproduct $\bigoplus_{i\in I}A_{i}$ with the canonical inclusions $\left\{\mu_{i}:A_{i}\rightarrow\bigoplus_{i\in I}A_{i}\right\}_{i\in I}$, and a set of exact sequences $\left\{\eta_{i}:\;0\rightarrow B\overset{f_{i}}{\rightarrow}E_{i}\overset{g_{i}}{\rightarrow}A_{i}\rightarrow 0\right\}_{i\in I}$. If there is an exact sequence $\eta:\>0\rightarrow B\overset{f^{\prime}}{\rightarrow}E\overset{g^{\prime}}{\rightarrow}\bigoplus_{i\in I}A_{i}\rightarrow 0$ such that $\overline{\eta}\mu_{i}=\overline{\eta_{i}}\,\forall i\in I$, then the morphism $f$ in the exact sequence

[TABLE]

is a monomorphism and $\overline{\eta}=\overline{\Theta(\eta_{i})}$.

Proof.

Consider the direct system $\left\{f_{i}:B\rightarrow E_{i}\right\}_{i\in I}$. We know that $\overline{\eta}\mu_{i}=\overline{\eta_{i}}\,\forall i\in I$. Hence, for every $i\in I$ there is a morphism of exact sequences

$(1,\nu_{i},\mu_{i}):\eta_{i}\rightarrow\eta\mbox{.}$

Observe that the set of morphisms $\left\{\nu_{i}:E_{i}\rightarrow E\right\}_{i\in I}$, together with the morphism $f^{\prime}:B\rightarrow E$, is a co-compatible family of morphisms.

[math]B$$E_{i}$$A_{i}[math][math]B$$E$$\bigoplus A_{i}[math]f_{i}$$g_{i}$$f^{\prime}$$g^{\prime}$$\nu_{i}$$\mu_{i}

Therefore, there is a unique morphism $\omega:\operatorname{colim}f_{i}\rightarrow E$ such that $\omega\sigma_{i}=\nu_{i}$ for every $i$ in $I$ and $\omega f=f^{\prime}$, where $\left\{\sigma_{i}:E_{i}\rightarrow\operatorname{colim}f_{i}\right\}_{i\in I}\cup\{f:B\rightarrow\operatorname{colim}f_{i}\}$ is

[math]B$$E_{i}$$A_{i}[math]B$$\operatorname{colim}(f_{i})$$\bigoplus A_{i}[math]f_{i}$$g_{i}$$f$$g$$\sigma_{i}$$\mu_{i}

the co-compatible family associated to the colimit. Notice that $\omega f=f^{\prime}$ is a monomorphism, so $f$ is also a monomorphism.

It remains to prove that $\overline{\eta}=\overline{\Theta(\eta_{i})}$. Observe that, by the cokernel universal property, we can build a morphism of exact sequences

$(1,\omega,\omega^{\prime}):\Theta(\eta_{i})\rightarrow\eta\mbox{.}$

It is enough to show that $\omega^{\prime}=1$. With that goal, we see that

$\omega^{\prime}\mu_{i}g_{i}=\omega^{\prime}g\sigma_{i}=g^{\prime}\omega\sigma_{i}=g^{\prime}\nu_{i}=\mu_{i}g_{i}\mbox{.}$

[math]B$$\operatorname{colim}(f_{i})$$\bigoplus A_{i}[math][math]B$$E$$\bigoplus A_{i}[math]f$$g$$f^{\prime}$$g^{\prime}$$\omega$$\omega^{\prime}

Hence, by the fact that $g_{i}$ is an epimorphism, $\omega^{\prime}\mu_{i}=\mu_{i}\,\forall i\in I$. Then, by the universal coproduct property we can conclude that $\omega^{\prime}=1$.

∎

Corollary 4.2**.**

Let $\mathcal{C}$ be an AB3 abelian category, $\left\{A_{i}\right\}_{i\in I}$ be a set of objects in $\mathcal{C}$, and $B\in\mathcal{C}$. Consider the coproduct $\left\{\mu_{i}:A_{i}\rightarrow\bigoplus_{i\in I}A_{i}\right\}_{i\in I}$, and the correspondence $\Psi_{1}:\operatorname{Ext}_{\mathcal{C}}^{1}\left(\bigoplus A_{i},B\right)\rightarrow\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{1}\left(A_{i},B\right)$. If $\Psi_{1}$ is biyective, then the inverse correspondence maps each $(\overline{\eta_{i}})\in\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{1}\left(A_{i},B\right)$, with representatives

[TABLE]

to the extension given by the exact sequence

[TABLE]

Theorem 4.3**.**

Let $\mathcal{C}$ be an Ab3 category. Then, $\mathcal{C}$ is an Ab4 category if, and only if, the correspondence $\Psi_{1}:\operatorname{Ext}_{\mathcal{C}}^{1}\left(\bigoplus X_{i},Y\right)\rightarrow\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{1}\left(X_{i},Y\right)$ is biyective for every $Y\in\mathcal{C}$ and every set of objects $\left\{X_{i}\right\}_{i\in I}$.

Proof.

By 3.11, it is enough to prove that if $\Psi$ is biyective for every $Y\in\mathcal{C}$ and every set of objects $\left\{X_{i}\right\}_{i\in I}$, then $\mathcal{C}$ is Ab4. With this purpose, we will consider a set of exact sequences $\left\{\eta_{i}:\>0\rightarrow A_{i}\overset{\alpha_{i}}{\rightarrow}B_{i}\overset{\beta_{i}}{\rightarrow}C_{i}\rightarrow 0\right\}_{i\in I}$ and prove that the morphism $\bigoplus_{i\in I}\alpha_{i}:\bigoplus_{i\in I}A_{i}\rightarrow\bigoplus_{i\in I}B_{i}$ is a monomorphism. Consider the coproduct $\bigoplus_{i\in I}A_{i}$ and the canonic inclusions $\mu_{i}:A_{i}\rightarrow\bigoplus_{i\in I}A_{i}$.

By the dual result of 2.7, for every $i$ in $I$ we have an exact sequence morphism $(\mu_{i},\mu_{i}^{\prime},1):\eta_{i}\rightarrow\mu_{i}\eta_{i}$, where

$\mu_{i}\eta_{i}:\>0\rightarrow\bigoplus_{i\in I}A_{i}\overset{f_{i}}{\rightarrow}E_{i}\overset{}{\rightarrow}C_{i}\rightarrow 0\mbox{.}$

Consider the correspondence

[math]A_{i}$$B_{i}$$C_{i}[math][math]\bigoplus_{i\in I}A_{i}$$E_{i}$$C_{i}[math]\scriptstyle\alpha_{i}$$\scriptstyle\beta_{i}$$\scriptstyle f_{i}$$\scriptstyle\mu_{i}$$\scriptstyle\mu_{i}^{\prime}

[TABLE]

By 4.2, we know that $\Psi^{-1}$ maps $(\mu_{i}\overline{\eta_{i}})\in\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{1}\left(C_{i},\bigoplus_{i\in I}A_{i}\right)$ to the extension given by the exact sequence

[TABLE]

A_{i}$$B_{i}$$\bigoplus A_{i}$$E_{i}$$X$$\alpha_{i}$$\mu^{\prime}_{i}$$\mu_{i}$$f_{i}$$\alpha$$g_{i}$$\gamma_{i}

We will show that $\operatorname{colim}f_{i}=\bigoplus_{i\in I}B_{i}$ to conclude that $\bigoplus_{i\in I}\alpha_{i}$ is a monomorphism. Indeed, consider a family of morphisms $\left\{g_{i}:B_{i}\rightarrow X\right\}_{i\in I}$. By the universal property of the coproduct $\bigoplus_{i\in I}A_{i}$, there is a unique morphism $\alpha:\bigoplus_{i\in I}A_{i}\rightarrow X$ such that $g_{i}\alpha_{i}=\alpha\mu_{i}\forall i\in I$. Now, by the universal property of the pushout on the last equality, for every $i\in I$ there is a unique morphism $\gamma_{i}:E_{i}\rightarrow X$ such that $g_{i}=\gamma_{i}\mu_{i}^{\prime}$ and $\alpha=\gamma_{i}f_{i}$.

Before going further, consider the co-compatible family of morphisms associated to the colimit $\left\{u_{k}:E_{k}\rightarrow\operatorname{colim}f_{i}\right\}_{k\in I}$. Observe that, by the universal property of the colimit on the last equalities, there is a unique morphism $\Lambda:\operatorname{colim}f_{i}\rightarrow X$ such that $\Lambda u_{i}=\gamma_{i}\,\forall i\in I$ and $\Lambda f=\alpha$.

In particular, if $X=\bigoplus_{i\in I}B_{i}$ and $\left\{g_{i}:B_{i}\rightarrow\bigoplus_{i\in I}B_{i}\right\}_{i\in I}$ is the set of canonic inclusions, there is a unique morphism $\Lambda:\operatorname{colim}f_{i}\rightarrow\bigoplus_{i\in I}B_{i}$ such that $\Lambda u_{i}=\gamma_{i}\,\forall i\in I$ and $\Lambda f=\alpha$. Furthermore, by the universal property of the coproduct $\bigoplus_{i\in I}B_{i}$, there is a unique morphism $\Lambda^{\prime}:\bigoplus_{i\in I}B_{i}\rightarrow\operatorname{colim}f_{i}$ such that $\Lambda^{\prime}g_{i}=u_{i}\mu^{\prime}_{i}\,\forall i\in I$.

We shall now prove that $\Lambda$ is an isomorphism and $\Lambda^{\prime}=\Lambda^{-1}$. Observe that

$\Lambda\Lambda^{\prime}g_{i}=\Lambda u_{i}\mu^{\prime}_{i}=\gamma_{i}\mu^{\prime}_{i}=g_{i}\,\forall i\in I\mbox{.}$

Hence, by the universal property of the coproduct $\bigoplus_{i\in I}B$, we can conclude that $\Lambda\Lambda^{\prime}=1$.

Next, we prove that $\Lambda^{\prime}\Lambda=1$. Observe that

A_{i}$$B_{i}$$\bigoplus A_{i}$$E_{i}$$\operatorname{colim}f_{i}$$X$$\operatorname{colim}f_{i}$$\alpha_{i}$$\mu^{\prime}_{i}$$\mu_{i}$$f_{i}$$f$$u_{i}$$\Lambda$$\Lambda^{\prime}$$\alpha$$g_{i}$$u_{i}\mu^{\prime}_{i}$$\gamma_{i}

[TABLE]

and also that

[TABLE]

Hence, by the last equalities and the universal property of the coproduct $\bigoplus_{i\in I}A_{i}$, we can conclude that $f=\Lambda^{\prime}\Lambda f$. Furthermore, observe that

[TABLE]

and also that

[TABLE]

Hence, by the last equalities and the universal property of the pushout $(E_{i},f_{i},\mu^{\prime}_{i})$, we can conclude that $\Lambda^{\prime}\Lambda u_{i}=u_{i}$. Now, it follows from the universal property of the colimit that $\Lambda^{\prime}\Lambda=1$. Therefore, $\Lambda$ is an isomorphism and $\Lambda^{\prime}=\Lambda^{-1}$.

By the last remark, without loss of generality $\operatorname{colim}f_{i}=\bigoplus_{i\in I}B_{i}$, $\Lambda=1=\Lambda^{\prime}$, and $g_{i}=u_{i}\mu^{\prime}_{i}\,\forall i\in I$. Now, observe that

[TABLE]

Hence, by the universal property of the corpoduct $\bigoplus_{i\in I}A_{i}$, we can conclude that $f=\bigoplus_{i\in I}\alpha_{i}$. Therefore, $\bigoplus_{i\in I}\alpha_{i}$ is a monomorphism. ∎

We have the following equivalences.

Theorem 4.4**.**

Let $\mathcal{C}$ be an Ab3 category. Then, the following statements are equivalent:

1. (a)

$\mathcal{C}$* is an Ab4 category.* 2. (b)

The correspondence $\Psi:\operatorname{Ext}_{\mathcal{C}}^{1}\left(\bigoplus_{i\in I}X_{i},Y\right)\rightarrow\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{1}\left(X_{i},Y\right)$ is biyective for every $Y\in\mathcal{C}$ and every set of objects $\left\{X_{i}\right\}_{i\in I}$. 3. (c)

The correspondence $\Psi_{n}:\operatorname{Ext}_{\mathcal{C}}^{n}\left(\bigoplus_{i\in I}X_{i},Y\right)\rightarrow\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{n}\left(X_{i},Y\right)$ is biyective $\forall Y\in\mathcal{C}$, every set of objects $\left\{X_{i}\right\}_{i\in I}$, and $\forall n>0$.

Proof.

It follows from 3.12 and 4.3. ∎

Remark 4.5*.*

For an example of an Ab3 category that is not Ab4, see the dual category of [9, Example A.4].

By duality we have the following result.

Theorem 4.6**.**

Let $\mathcal{C}$ be an Ab3 category. Then, the following statements are equivalent:*

1. (a)

$\mathcal{C}$* is an Ab4* category.* 2. (b)

The correspondence $\Psi:\operatorname{Ext}_{\mathcal{C}}^{1}\left(Y,\prod_{i\in I}X_{i}\right)\rightarrow\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{1}\left(Y,X_{i}\right)$ is biyective for every $Y\in\mathcal{C}$ and every set of objects $\left\{X_{i}\right\}_{i\in I}$. 3. (c)

The correspondence $\Psi_{n}:\operatorname{Ext}_{\mathcal{C}}^{n}\left(Y,\prod_{i\in I}X_{i}\right)\rightarrow\prod_{i\in I}\operatorname{Ext}_{\mathcal{C}}^{n}\left(Y,X_{i}\right)$ is biyective $\forall Y\in\mathcal{C}$, every set of objects $\left\{X_{i}\right\}_{i\in I}$, and $\forall n>0$.

Acknowledgements

I wish to thank my advisor Octavio Mendoza for encouraging me to publish this work and for proof reading the article. I am also grateful to Sergio Estrada whose comments greatly improved the quality of this paper.