  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.
