The negative side of cohomology for Calabi-Yau categories
  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. 
Petter Andreas Bergh, David A. Jorgensen and Steffen Oppermann
math.RA math.AC math.CT math.KT