Stable invariance of the restricted Lie algebra structure of Hochschild cohomology

Benjamin Briggs and Lleonard Rubio y Degrassi

Vol. 321 (2022), No. 1, 45–71

DOI: 10.2140/pjm.2022.321.45

Abstract

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 ${\mathrm{HH}<!--FUNCTION\; APPLICATION-->}^1(A,A)$. 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 $B_{\infty }$-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 $B_{\infty }$-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.

Keywords

Hochschild cohomology, Gerstenhaber bracket, restricted Lie algebra, B-infinity algebra, stable equivalence of Morita type, singularity category

Mathematical Subject Classification

Primary: 16E40, 16D90

Secondary: 17B50, 13D03

Milestones

Received: 25 April 2022

Revised: 8 September 2022

Accepted: 24 September 2022

Published: 3 March 2023