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 selfinjective algebras. Thereby we obtain a number of positive characteristic stable invariants, such as the p-toral rank of 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. We provide several applications to commutative algebra and modular representation theory. The proof exploits in an essential way the B∞-structure of the Hochschild cochain complex.In the appendix we explain how the well-definedness of the p-power operation on Hochschild cohomology follows from some (originally topological) results of May and Cohen, and (on the algebraic side) Turchin. We give complete proofs, using the language of operads.

Maximal tori in $HH^1$ and the fundamental group

“…This passes to the quotient Der(A)/ Inn(A) to make HH 1 (A) a restricted Lie algebra as well, cf. [16]. Definition 3.11.…”

“…Similarly, among self-injective algebras Out(A) • is invariant under stable equivalences of Morita type [49,Théorème 4.3], so mt-rank(HH 1 (A)) = mt-rank L(Out(A)) is as well in characteristic zero, and in positive characteristic the desired statement follows from [16,Theorem 1].…”