Tate-Hochschild homology and cohomology of Frobenius algebras

TL;DR

This paper investigates Tate-Hochschild (co)homology for Gorenstein algebras, establishing duality theorems for Frobenius algebras and computing these groups explicitly for specific quantum complete intersections.

Contribution

It introduces a comprehensive framework for Tate-Hochschild (co)homology in all degrees and proves duality theorems for Frobenius algebras, with explicit calculations for quantum complete intersections.

Findings

Tate-Hochschild (co)homology groups are defined for all degrees, including negative.

Duality theorems relate positive and negative degree groups for Frobenius algebras.

Explicit computations are provided for certain quantum complete intersections.

Abstract

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