The negative side of cohomology for Calabi-Yau categories

TL;DR

This paper investigates the structure of integer-graded cohomology rings in Calabi-Yau categories, revealing that negative degree cohomology is trivial under certain conditions, with implications for Tate cohomology in algebraic contexts.

Contribution

It establishes a general result about the triviality of negative degree cohomology in Calabi-Yau categories when the non-negative part admits a specific regular sequence.

Findings

Negative degree cohomology is a trivial extension of non-negative degree cohomology.

Product of elements in negative degrees is zero.

Results apply to Tate-Hochschild cohomology and Tate cohomology over group algebras and Gorenstein rings.

Abstract

We study integer-graded cohomology rings defined over Calabi-Yau categories. We show that the cohomology in negative degree is a trivial extension of the cohomology ring in non-negative degree, provided the latter admits a regular sequence of central elements of length two. In particular, the product of elements of negative degrees are zero. As corollaries we apply this to Tate-Hochschild cohomology rings of symmetric algebras, and to Tate cohomology rings over group algebras. We also prove similar results for Tate cohomology rings over commutative local Gorenstein rings.