Local coefficients for genuine equivariant cohomology I
Abstract
We develop a theory of equivariant local systems which categorifies genuine equivariant cohomology theories -- such as Atiyah--Segal K-theory, Lurie's tempered cohomology, and the equivariant elliptic cohomology of Grojnowski, Greenlees, and Gepner--Meier -- analogously to how the category of ordinary local systems categorifies singular cohomology. More precisely, we introduce, for a global space X and a coefficient system 𝒜 : Orbop → PrL, the categories LSglo(X,𝒜) and LSgen(X,𝒜) of globally equivariant and genuine local systems, the latter generalizing the genuine stable category SpG of a compact Lie group G to non-constant coefficients. When the coefficients come from an oriented abelian group stack A over a locally complex periodic base, we also construct the category LStemp(X,A) of tempered local systems, extending Lurie's theory beyond finite groups; the defining condition is justified by a tempered form of the Atiyah--Segal completion theorem. We show that global sections induce an equivalence LStemp(BU(1),A) ≃ QCoh(A) and we prove that LStemp(X,A) is a smashing localization of LSgen(X,A) if X is an orbispace.
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BibTeX
@misc{konovalov2026local,
title = {{Local coefficients for genuine equivariant cohomology I}},
author = {Nikolai Konovalov and Artem Prikhodko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23749},
eprint = {2609.23749},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23749},
}
RIS
TY - UNPB
TI - Local coefficients for genuine equivariant cohomology I
AU - Nikolai Konovalov
AU - Artem Prikhodko
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23749
UR - https://arxiv.org/abs/2609.23749
AB - We develop a theory of equivariant local systems which categorifies genuine equivariant cohomology theories -- such as Atiyah--Segal K-theory, Lurie's tempered cohomology, and the equivariant elliptic cohomology of Grojnowski, Greenlees, and Gepner--Meier -- analogously to how the category of ordinary local systems categorifies singular cohomology. More precisely, we introduce, for a global space X and a coefficient system 𝒜 : Orb^op → Pr^L, the categories LS^glo(X,𝒜) and LS^gen(X,𝒜) of globally equivariant and genuine local systems, the latter generalizing the genuine stable category Sp^G of a compact Lie group G to non-constant coefficients. When the coefficients come from an oriented abelian group stack A over a locally complex periodic base, we also construct the category LS^temp(X,A) of tempered local systems, extending Lurie's theory beyond finite groups; the defining condition is justified by a tempered form of the Atiyah--Segal completion theorem. We show that global sections induce an equivalence LS^temp(BU(1),A) ≃ QCoh(A) and we prove that LS^temp(X,A) is a smashing localization of LS^gen(X,A) if X is an orbispace.
ER -