Lemma 45.7.7. Let H^* be a classical Weil cohomology theory (Definition 45.7.3). Let X be a smooth projective variety of dimension d. Choose a basis e_{i, j}, j = 1, \ldots , \beta _ i of H^ i(X) over F. Using Künneth write
\gamma ([\Delta ]) = \sum \nolimits _{i = 0, \ldots , 2d} \sum \nolimits _ j e_{i, j} \otimes e'_{2d - i , j} \quad \text{in}\quad \bigoplus \nolimits _ i H^ i(X) \otimes _ F H^{2d - i}(X)
with e'_{2d - i, j} \in H^{2d - i}(X). Then \int _ X e_{i, j} \cup e'_{2d - i, j'} = (-1)^ i\delta _{jj'}.
Proof.
Recall that \Delta ^* : H^*(X \times X) \to H^*(X) is equal to the cup product map H^*(X) \otimes _ F H^*(X) \to H^*(X), see Remark 45.7.2. On the other hand we have \gamma ([\Delta ]) = \Delta _*\gamma ([X]) = \Delta _*1 by axiom (C)(b) and the fact that \gamma ([X]) = 1. Namely, [X] \cdot [X] = [X] hence by axiom (C)(c) the cohomology class \gamma ([X]) is 0 or 1 in the 1-dimensional F-algebra H^0(X); here we have also used axioms (A)(d) and (A)(b). But \gamma ([X]) cannot be zero as [X] \cdot [x] = [x] for a closed point x of X and we have the nonvanishing of \gamma ([x]) by Lemma 45.7.4. Hence
\int _{X \times X} \gamma ([\Delta ]) \cup a \otimes b = \int _{X \times X} \Delta _*1 \cup a \otimes b = \int _ X a \cup b
by the definition of \Delta _*. On the other hand, we have
\int _{X \times X} (\sum e_{i, j} \otimes e'_{2d -i , j}) \cup a \otimes b = \sum (\int _ X a \cup e_{i, j})(\int _ X e'_{2d - i, j} \cup b)
by Lemma 45.7.5; note that we made two switches of order so that the sign is 1. Thus if we choose a such that \int _ X a \cup e_{i, j} = 1 and all other pairings equal to zero, then we conclude that \int _ X e'_{2d - i, j} \cup b = \int _ X a \cup b for all b, i.e., e'_{2d - i, j} = a. This proves the lemma.
\square
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