Homological finiteness of functors on an additive category and applications

TL;DR

This paper establishes conditions for finite-length functors on additive categories to have finitely generated projective resolutions, explores homological finiteness for polynomial functors, and applies these results to matrix monoids and linear groups.

Contribution

It introduces new criteria for homological finiteness of functors and extends existing theories to broader contexts, including twisted stability and stable homology.

Findings

Finite-length functors have finitely generated projective resolutions under certain conditions.

Homological finiteness properties are established for polynomial functors.

Applications include results on twisted homological stability and stable homology of linear groups.

Abstract

We give sufficient conditions which ensure that a functor of finite length from an additive category to finite-dimensional vector spaces has a projective resolution whose terms are finitely generated. For polynomial functors, we study also a weaker homological finiteness property, which applies to twisted homological stability for matrix monoids. This is inspired by works by Schwartz and Betley-Pirashvili, which are generalised; this also uses decompositions {\`a} la Steinberg over an additive category that we recently got with Vespa. We show also, as an application, a finiteness property for stable homology of linear groups on suitable rings.