Abstract
Abstract
There are well-known identities involving the Ext bifunctor, coproducts, and products in AB4 abelian categories with enough projectives. Namely, for every such category
\[\mathcal{A}\]
, given an object X and a set of objects
\[{\{ {{\text{A}}_{\text{i}}}\} _{{\text{i}} \in {\text{I}}}}\]
, an isomorphism
\[Ext_\mathcal{A}^{\text{n}}({ \oplus _{{\text{i}} \in {\text{I}}}}{{\text{A}}_{\text{i}}},{\text{X}}) \cong \prod\nolimits_{{\text{i}} \in {\text{I}}} {Ext_\mathcal{A}^{\text{n}}({{\text{A}}_{\text{i}}},{\text{X}})} \]
can be built, where
\[Ex{t^{\text{n}}}\]
is the nth derived functor of the Hom functor. The goal of this paper is to show a similar isomorphism for the nth Yoneda Ext, which is a functor equivalent to
\[Ex{t^{\text{n}}}\]
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 projectives nor injectives, extending a result by Colpi and Fuller in [8]. Furthermore, the isomorphisms constructed are used to characterize AB4 categories. A dual result is also stated.
Publisher
Cambridge University Press (CUP)
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