Hochschild cohomology of twisted tensor products

Abstract

The tensor product $$R\otimes S$$ of two algebras can have its multiplication deformed by a bicharacter to yield a twisted tensor product $$R\otimes ^t S$$ . We completely describe the Hochschild cohomology of $$R\otimes ^t S$$ in terms of the Hochschild cohomology of the components R and S, including the full Gerstenhaber algebra structure. This description generalizes a result of Bergh and Oppermann. A number of interesting classes of noncommutative algebras arise as bicharacter twisted tensor products, sometimes in non-obvious ways. The main result thereby allows us to significantly simplify various calculations in the literature, and to compute Hochschild cohomology in several new classes of examples. In particular, we fully compute the Hochschild cohomology of quantum complete intersection algebras, with any number of indeterminates. One new tool which goes into the main theorem is orbit Hochschild cohomology, which can be defined for algebras with a group action, and which satisfies twisted versions of the usual Gerstenhaber algebra axioms.