Lemma 45.7.6 (0FGY)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 45: Weil Cohomology Theories
Section 45.7: Classical Weil cohomology theories
Lemma 45.7.6 (cite)

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Lemma 45.7.6. Let $H^*$ be a classical Weil cohomology theory (Definition 45.7.3). Let $X$ and $Y$ be smooth projective varieties. Then $\text{pr}_{2, *} : H^*(X \times Y) \to H^*(Y)$ sends $a \otimes b$ to $(\int _ X a) b$.

Proof. This is equivalent to the result of Lemma 45.7.5. $\square$

Comments (2)

Comment #7510 by Hao Peng on July 24, 2022 at 17:15

This one is jot trivial though. You need to use Poincare duality to deduce it from tag 0FGX.

Comment #7646 by Stacks Project on August 14, 2022 at 12:43

It is indeed the case that sometimes the proofs in the Stacks project aren't trivial and the reader has to think a little bit. In this case I think the statement more or less tells you what to try.

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