Invariant and Smash-Product Hochschild Cohomology

Definitions and mathematical statement

The supplied question omits hypotheses on its algebraic data. The following gives a conventional framework for interpreting its notation; these hypotheses are not independently confirmed by the supplied record.

Let $k$ be a field of characteristic $p>0$, let $G$ be a finite group with $p\mid |G|$, and let $A$ be an associative unital $k$-algebra equipped with an action of $G$ by $k$-algebra automorphisms. Define the smash product $B=A\# kG$ to have underlying vector space $A\otimes_k kG$, with multiplication \[ (a\# g)(b\# h)=a\,g(b)\# gh \qquad(a,b\in A,\ g,h\in G). \] Suppose that $M$ is a $B$-bimodule, and regard it as an $A$-bimodule by restriction along $a\mapsto a\#1$. For an algebra $C$ and a $C$-bimodule $N$, write \[ C^e=C\otimes_k C^{\mathrm{op}},\qquad \operatorname{HH}^n(C,N)=\operatorname{Ext}_{C^e}^n(C,N),\qquad \operatorname{HH}^*(C,N)=\bigoplus_{n\geq0}\operatorname{HH}^n(C,N). \] Writing $u_g=1_A\#g$, the action $g\cdot m=u_g m u_{g^{-1}}$, together with the action on $A$, induces the usual action on Hochschild cohomology. In cochain degree $n$, this action is represented by \[ (g\cdot f)(a_1,\ldots,a_n) =u_g f\left(g^{-1}(a_1),\ldots,g^{-1}(a_n)\right)u_{g^{-1}}. \] Define \[ \operatorname{HH}^*(A,M)^G =\{x\in\operatorname{HH}^*(A,M):g\cdot x=x\text{ for every }g\in G\}. \]

Source question, with unresolved qualifications. In the modular case $p\mid |G|$, determine the relationship between \[ \operatorname{HH}^*(A,M)^G \quad\text{and}\quad \operatorname{HH}^*(A\# kG,M). \] The source calls these objects graded algebras. For an arbitrary bimodule $M$, however, the definitions above supply graded vector spaces without a specified internal cup product. Suitable compatible multiplicative structure on $M$, or an intended specialization of $M$, must be supplied before the graded-algebra formulation is determined. The source does not specify this structure, the full class of admissible $k,G,A,M$, or a particular comparison map, identity, inequality, or other relationship to be established. Accordingly, this is an unresolved comparison question in a conditional framework, not a fully specified conjectural assertion.

Short English statement

For a finite group acting on an algebra over a field whose characteristic divides the group order, compare the invariant Hochschild cohomology of the algebra with the Hochschild cohomology of its smash product. The source leaves the coefficient object $M$, the structure needed to obtain graded algebras, and the precise intended comparison insufficiently specified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003061$, title Invariant and Smash-Product Hochschild Cohomology; exact source problem code catalogue entry. The record places the problem in the Oberwolfach Reports Open Problems collection and gives the proposed year as 2016; no individual proposer is supplied.
• Supplied report citation: Hochschild Cohomology in Algebra, Geometry, and Topology (2016), Oberwolfach Reports, pp. 449--506, DOI 10.4171/owr/2016/10. https://doi.org/10.4171/owr/2016/10. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or named contributor is supplied.
• Supplied literature source, labelled "OWR report": https://doi.org/10.4171/owr/2016/10.
• Supplied literature source, labelled "smash products in the nonsemisimple case": https://arxiv.org/abs/1511.05388.
• Supplied literature source, labelled "smash-product spectral sequences": https://arxiv.org/abs/1401.3551.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that a modular spectral sequence gives inequalities and collapse criteria for cyclic $p$-groups and certain extensions, while the general relation between invariant Hochschild cohomology and smash-product Hochschild cohomology remains unresolved. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that mathematical content was preserved while notation, typography, or omitted definitions were repaired. This is supplied registry metadata, not independent mathematical verification of this draft.