Invariance of the restricted $p$-power map on integrable derivations under stable equivalences

TL;DR

This paper proves that the $p$-power map on integrable derivations in Hochschild cohomology remains invariant under stable equivalences of Morita type for certain algebras, highlighting a specific invariance property.

Contribution

It establishes the invariance of the $p$-power map on integrable derivations under stable equivalences of Morita type for selfinjective algebras, and provides a counterexample for general transfer maps.

Findings

$p$-power maps commute with stable equivalences of Morita type on integrable derivations.

Counterexample shows $p$-power maps do not always commute with transfer maps.

Invariance holds in the context of selfinjective algebras over fields of prime characteristic.

Abstract

We show that the $p$-power maps in the first Hochschild cohomology space of finite-dimensional selfinjective algebras over a field of prime characteristic $p$ commute with stable equivalences of Morita type on the subgroup of classes represented by integrable derivations. We show, by giving an example, that the $p$-power maps do not necessarily commute with arbitrary transfer maps in the Hochschild cohomology of symmetric algebras.