# Tate-Hochschild homology and cohomology of Frobenius algebras

### Petter Andreas Bergh

NTNU Trondheim, Norway### David A. Jorgensen

University of Texas at Arlington, USA

## 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.

## Cite this article

Petter Andreas Bergh, David A. Jorgensen, Tate-Hochschild homology and cohomology of Frobenius algebras. J. Noncommut. Geom. 7 (2013), no. 4, pp. 907–937

DOI 10.4171/JNCG/139