On direct summands of homological functors on length categories

TL;DR

This paper proves that certain additive functors in length categories are closed under direct summands, confirming a conjecture of Auslander for categories of finite modules over artin algebras and characterizing injectives among finitely presented functors.

Contribution

It establishes that direct summands of specific bifunctors retain their form under conditions like finite length or chain conditions, confirming Auslander's conjecture in key cases.

Findings

Direct summands of bifunctors are of the same form under certain conditions.

Covariant Ext functors are the only injectives among defect-zero finitely presented functors.

Positive resolution of Auslander's conjecture for categories of finite modules over artin algebras.

Abstract

We show that direct summands of certain additive functors arising as bifunctors with a fixed argument in an abelian category are again of that form whenever the fixed argument has finite length or, more generally, satisfies the descending chain condition on images of nested endomorphisms. In particular, this provides a positive answer to a conjecture of M. Auslander in the case of categories of finite modules over artin algebras. This implies that the covariant Ext functors are the only injectives in the category of defect-zero finitely presented functors on such categories.