I'm interested in computing the ring structure on the Tate cohomology $\hat{H}^*(C_2 \times C_2, \mathbb{F}_2)$. It's easy enough to compute the ring structure on nonnegative degrees, along with the group structure in general, but I don't know how to extend the cup product to negative degrees. Is the answer known?
2026-03-30 07:43:58.1774856638
What is $\hat{H}^*(C_2 \times C_2, \mathbb{F}_2)$ as a ring?
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Answering for completeness. John Palmieri pointed me at another MathOverflow question which answers what I'm looking for. See https://web.archive.org/web/20130630192620/http://www.math.uwo.ca/~schebolu/research/Jan/tateprop.pdf