Almost split sequences and approximations

TL;DR

This paper investigates conditions under which almost split sequences exist in exact categories and their subcategories, providing criteria and applications for their transfer and existence.

Contribution

It establishes necessary and sufficient conditions for the existence of almost split sequences in exact categories and subcategories, extending the understanding of their structure and transfer.

Findings

Criteria for almost split sequences in exact categories

Conditions for inducing almost split sequences in subcategories

Applications demonstrating the theoretical results

Abstract

Let A be an exact category, that is, an extension-closed full subcategory of an abelian category. Firstly, we give some necessary and sufficient conditions for A to have almost split sequences. Then, we study when an almost split sequence in A induces an almost split sequence in an exact subcategory C of A. In case A has almost split sequences and C is Hom-finite Krull-Schmidt, this provides a necessary and sufficient condition for C to have almost split sequences. Finally, we show two applications of these results.