Stable invariance of the restricted Lie algebra structure of Hochschild cohomology
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We show that the restricted Lie algebra structure on Hochschild cohomology is invariant under stable equivalences of Morita type between self-injective algebras. Thereby, we obtain a number of positive characteristic stable invariants, such as the p-toral rank of HH1 (A, A). We also prove a more general result concerning Iwanaga–Gorenstein algebras, using a generalization of stable equivalences of Morita type. Several applications are given to commutative algebra and modular representation theory. These results are proven by first establishing the stable invariance of the B∞-structure of the Hochschild cochain complex. In the appendix, we explain how the p-power operation on Hochschild cohomology can be seen as an artifact of this B∞-structure. In particular, we establish well-definedness of the p-power operation, following some—originally topological—methods due to May, Cohen and Turchin, using the language of operads.
Originalsprog | Engelsk |
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Tidsskrift | Pacific Journal of Mathematics |
Vol/bind | 321 |
Udgave nummer | 1 |
Sider (fra-til) | 45-73 |
Antal sider | 29 |
ISSN | 0030-8730 |
DOI | |
Status | Udgivet - 2022 |
Bibliografisk note
Funding Information:
Rubio y Degrassi was supported by the Fundación “Séneca” of Murcia (19880/GERM/15) and by an INdAM postdoctoral research grant (2019-2020). MSC2020: primary 16E40, 16D90; secondary 17B50, 13D03. Keywords: Hochschild cohomology, Gerstenhaber bracket, restricted Lie algebra, B-infinity algebra, stable equivalence of Morita type, singularity category.
Publisher Copyright:
© 2022 Mathematical Sciences Publishers
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