Lemma 45.7.5. Let $H^*$ be a classical Weil cohomology theory (Definition 45.7.3). Let $X$ and $Y$ be smooth projective varieties. Then $\int _{X \times Y} = \int _ X \otimes \int _ Y$.
Proof. Say $\dim (X) = d$ and $\dim (Y) = e$. By axiom (B) we have $H^{2d + 2e}(X \times Y) = H^{2d}(X) \otimes H^{2e}(Y)$ and by axiom (A)(d) this is $1$-dimensional. By Lemma 45.7.4 this $1$-dimensional vector space generated by the class $\gamma ([x \times y])$ of a closed point $(x, y)$ and $\int _{X \times Y} \gamma ([x \times y]) = 1$. Since $\gamma ([x \times y]) = \gamma ([x]) \otimes \gamma ([y])$ by axioms (C)(a) and (C)(c) and since $\int _ X \gamma ([x]) = 1$ and $\int _ Y \gamma ([y]) = 1$ we conclude. $\square$
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