The Stacks project

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 K√ľnneth 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. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 45.7: Classical Weil cohomology theories

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FGZ. Beware of the difference between the letter 'O' and the digit '0'.