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

On the Lie algebra structure of integrable derivations

“…Indeed, this time one computes $$$$\begin{equation*} \alpha ^p = \operatorname{id}+ \alpha _m^pt^{pm}+\cdots \end{equation*}$$$$for $$\alpha \in \operatorname{Aut}_m(A[t]/(t^n))$$. This structure is studied in [26]. This remark shows that $$[m,n)$$‐integrable derivations arise even if one is interested only in $$[1,n)$$‐integrable derivations.…”

“…In positive characteristic this respects the $$p$$‐power structure by [15, (3.2)] combined with [5, Theorem 2]. For the case of self‐injective algebras that are stably equivalent of Morita type, there is by [18, Theorem 5.1] an isomorphism $${\operatorname{HH}}_{\rm int}^1(A)\cong {\operatorname{HH}}_{\rm int}^1(B)$$ induced by a transfer map, and this is an isomorphism of restricted Lie algebras by Corollary 1 together with [26, Theorem 1.1] and [5, Theorem 1].$$\Box$$…”

On blocks of defect two and one simple module, and Lie algebra structure of HH1

“…For example, in [1] it is assumed that q is not a root of unity. The last statement in Theorem 1.1 regarding the p-restricted structure of L is motivated by invariance results of p-power maps in Hochschild cohomology under derived and stable equivalences in work of Zimmermann [23] and Rubio y Degrassi [18].…”

Stable invariance of the restricted Lie algebra structure of Hochschild cohomology

“…The invariance of this structure is delicate because the p-power structure is non-linear, and therefore difficult to handle functorially. A partial answer was given by the second author in [36] for the subclass of integrable derivations. We answer Linckelmann's question affirmatively: Theorem 1.…”