# 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.

## Key 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.