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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