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The negative side of cohomology for Calabi–Yau categories

We study Z ‐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 2. In particular, the products 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.