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Hochschild cohomology and group actions

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Abstract

Given a finite group action on a (suitably enhanced) triangulated category linear over a field, we establish a formula for the Hochschild cohomology of the category of invariants, assuming the order of the group is coprime to the characteristic of the base field. The formula shows that the cohomology splits canonically with one summand given by the invariant subspace of the Hochschild cohomology of the original category. We also prove that Serre functors act trivially on Hochschild cohomology, and combine this with our formula to give a useful mechanism for computing the Hochschild cohomology of fractional Calabi–Yau categories.

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Acknowledgements

This paper greatly benefitted from discussions with Alexander Efimov, Valery Lunts, Jacob Lurie, and especially Akhil Mathew. I would also like to heartily thank Sasha Kuznetsov, for inspiring conversations during which many of the ideas in this paper were discovered. The original impetus for this paper was the application to GM varieties in our joint work [19]. Finally, I thank the referee for several useful comments.

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  1. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Alexander Perry

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  1. Alexander Perry
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Correspondence to Alexander Perry.

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This work was partially supported by NSF postdoctoral fellowship DMS-1606460, NSF Grant DMS-1902060, and the Institute for Advanced Study.

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Perry, A. Hochschild cohomology and group actions. Math. Z. 297, 1273–1292 (2021). https://doi.org/10.1007/s00209-020-02557-x

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