The Yoneda Ext and arbitrary coproducts in abelian categories

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 ${\mathrm{Ext}}^n({⨁}_{i\in I}{A}_i,X)\cong {\prod }_{i\in I}{\mathrm{Ext}}^n(A_i,X)$ and ${\mathrm{Ext}}^n(X,{\prod }_{i\in I}{A}_i)\cong {\prod }_{i\in I}{\mathrm{Ext}}^n(X,A_i)$ 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.

20 pages