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Bounded functor cohomology and comparison theorem
Abstract
Let k be a finite field. We consider the category of all functors ℱn from the category of k-vector spaces of dimension ≤ n into the category of all k-vector spaces. We compare its derived category to the derived category of representations of the general linear group GLn(k) using the extension by zero functor procedure. We upgrade the well-known recollement of abelian categories into the recollement of the corresponding derived categories. Then we obtain some translations of the cohomology the general linear group over finite field. We also compare ℱn with the category of polynomial functors 𝒫d,n, giving further reformulations over big finite fields.
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BibTeX
@misc{janowicz2026bounded,
title = {{Bounded functor cohomology and comparison theorem}},
author = {Karol Janowicz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34664},
eprint = {2609.34664},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34664},
}
RIS
TY - UNPB
TI - Bounded functor cohomology and comparison theorem
AU - Karol Janowicz
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34664
UR - https://arxiv.org/abs/2609.34664
AB - Let k be a finite field. We consider the category of all functors ℱ_n from the category of k-vector spaces of dimension ≤ n into the category of all k-vector spaces. We compare its derived category to the derived category of representations of the general linear group GL_n(k) using the extension by zero functor procedure. We upgrade the well-known recollement of abelian categories into the recollement of the corresponding derived categories. Then we obtain some translations of the cohomology the general linear group over finite field. We also compare ℱ_n with the category of polynomial functors 𝒫_d,n, giving further reformulations over big finite fields.
ER -