Stable invariance of the restricted Lie algebra structure of Hochschild cohomology

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 ${\mathrm{HH}}^1(A,A)$. We also prove a more general result concerning Iwanaga-Gorenstein algebras, using a more general notion 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.

arXiv comment20 pages, v3: small changes in the abstract and in the introduction

References in corpus (7)

• Triangulated categories of singularities and D-branes in Landau-Ginzburg models
• Operads, homotopy algebra and iterated integrals for double loop spaces
• Homotopy G-algebras and moduli space operad
• Dyer-Lashof-Cohen operations in Hochschild cohomology
• On blocks stably equivalent to a quantum complete intersection of dimension 9 in characteristic 3 and a case of the abelian defect group conjecture
• On the BV structure of the Hochschild cohomology of finite group algebras
• On the first Hochschild cohomology of cocommutative Hopf algebras of finite representation type

Cited by in corpus (2)

• Leavitt path algebras, $B_{\infty }$-algebras and Keller's conjecture for singular Hochschild cohomology
• Maximal tori in $H{H}^1$ and the fundamental group