Homological finiteness of functors on an additive category and applications

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 {à} 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.

in French language

References in corpus (4)

• Gröbner methods for representations of combinatorial categories
• Representation stability and finite linear groups
• Homology of FI-modules
• Connectedness of cup products for polynomial representations of $G{L}_n$ and applications

Cited by in corpus (1)

• Sur la noeth{é}rianit{é} locale des foncteurs polynomiaux