Tate-Hochschild homology and cohomology of Frobenius algebras

Abstract

Let $\Lambda$ be a two-sided Noetherian Gorenstein $k$-algebra, for $k$ a field. We introduce Tate–Hochschild homology and cohomology groups for $\Lambda$, 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 $\Lambda$. We prove certain duality theorems relating the Tate–Hochschild (co)homology groups in positive degree to those in negative degree, in the case where $\Lambda$ is a Frobenius algebra. We explicitly compute all Tate–Hochschild (co)homology groups for certain classes of Frobenius algebras, namely, certain quantum complete intersections.