Tate-Hochschild homology and cohomology of Frobenius algebras

Abstract

Let Λ be a two-sided Noetherian Gorenstein k-algebra, for k a field. We introduce Tate-Hochschild homology and cohomology groups for Λ, which are defined for all degrees, non-negative as well as negative, and which agree with the usual Hochschild homology and cohomology groups for all degrees larger than the injective dimension of Λ. We prove certain duality theorems relating the Tate-Hochschild (co)homology groups in positive degree to those in negative degree, in the case where Λ is a Frobenius algebra. We explicitly compute all Tate-Hochschild (co)homology groups for certain classes of Frobenius algebras, namely, certain quantum complete intersections.