Sur la structure des repr{é}sentations g{é}n{é}riques des groupes lin{é}aires infinis
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915072597557248 |
|---|---|
| author | Djament, Aurélien Touzé, Antoine |
| author_facet | Djament, Aurélien Touzé, Antoine |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15231 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sur la structure des repr{é}sentations g{é}n{é}riques des groupes lin{é}aires infinis Djament, Aurélien Touzé, Antoine Category Theory Algebraic Topology Representation Theory 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. |
| title | Sur la structure des repr{é}sentations g{é}n{é}riques des groupes lin{é}aires infinis |
| topic | Category Theory Algebraic Topology Representation Theory |
| url | https://arxiv.org/abs/2412.15231 |