The finite presentation of the stable Hom functors, the Bass torsion, and the cotorsion coradical

TL;DR

This paper characterizes when stable Hom functors are finitely presented, linking their defects to Bass torsion and cotorsion, and explores finite presentation conditions for tensor products with notable applications.

Contribution

It establishes necessary and sufficient conditions for stable Hom functors to be finitely presented and connects these to Bass torsion, cotorsion, and tensor product stabilization.

Findings

Stable Hom functors' defects relate to Bass torsion and cotorsion.

Conditions for finite presentation of tensor product sub-stabilization.

Application of finite presentation results to new contexts.

Abstract

We provide necessary and/or sufficient conditions for the stable Hom functors to be finitely presented. When the covariant Hom functor modulo projectives is finitely presented, its defect is isomorphic to the Bass torsion of the fixed argument. When the contravariant Hom functor modulo injectives is finitely presented, its defect is isomorphic to the cotorsion of the fixed argument. We also give a sufficient condition for the sub-stabilization of the tensor product to be finitely presented. A finite presentation of the tensor product leads to an unexpected application.