Sur la structure des repr{\'e}sentations g{\'e}n{\'e}riques des groupes lin{\'e}aires infinis

TL;DR

This paper investigates the structure of functor categories related to linear groups over rings, showing that finitely generated functors with finite-dimensional values can be decomposed into polynomial or reduced functors, ensuring they are noetherian.

Contribution

It introduces a framework for decomposing finitely generated functors into polynomial or reduced components, extending understanding of their algebraic properties.

Findings

Finitely generated functors are built from polynomial or reduced functors.

Such functors are always noetherian.

Finitely generated rings imply finitely generated projective resolutions.

Abstract

We study several structure aspects of functor categories from a small additive category to a module category, in particular the category F(A,K) of functors from finitely generated free modules over a commutative ring A to vector spaces over a field K -- such functors are sometimes called \textit{generic representations} of linear groups over A with coefficients in K. We are especially interested with finitely generated functors of F(A,K) taking finite dimensional values. We prove that they can, under a mild extra assumption (always satisfied if the ring A is noetherian), be built from much better understood functors, namely polynomial functors (in the sense of Eilenberg-MacLane), or factorising at the source through reduction modulo a cofinite ideal of A. We deduce that such functors are always noetherian et that, if the ring A is finitely generated, they have finitely generated projective resolutions.Our methods rely mainly on the study of weight decompositions of functors and their cross-effects, our recent previous work with Vespa (Ann. ENS 2023) and elementary commutative algebra.