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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