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Finite group actions on dg categories and Hochschild homology

Part of: Homological algebra Homological methods

Published online by Cambridge University Press:  17 March 2025

Ville Nordström
Ville Nordström*
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97401, United States
*
e-mail: villen@uoregon.edu
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Abstract

Let G be a finite group whose order is not divisible by the characteristic of the ground field $\mathbb {F}$. We prove a decomposition of the Hochschild homology groups of the equivariant dg category $\mathscr {C}^G$ associated with the action of G on a small dg category $\mathscr {C}$ which admits finite direct sums. When, in addition, the ground field $\mathbb {F}$ is algebraically closed this decomposition is related to a categorical action of $\text {Rep}(G)$ on $\mathscr {C}^G$ and the resulting action of the representation ring $R_{\mathbb {F}}(G)$ on $HH_\bullet (\mathscr {C}^G)$.

Keywords

Hochschild homologydg categoriesgroup actions

MSC classification

Primary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.) 18G35: Chain complexes

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 4 , December 2025 , pp. 1088 - 1108
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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