The Stacks project

Lemma 45.7.8. Let $H^*$ be a classical Weil cohomology theory (Definition 45.7.3). Let $X$ be a smooth projective variety. We have

\[ \sum \nolimits _{i = 0, \ldots , 2\dim (X)} (-1)^ i\dim _ F H^ i(X) = \deg ([\Delta ] \cdot [\Delta ]) = \deg (c_ d(\mathcal{T}_ X) \cap [X]) \]

Proof. Equality on the right. We have $[\Delta ] \cdot [\Delta ] = \Delta _*(\Delta ^![\Delta ])$ (Chow Homology, Lemma 42.62.6). Since $\Delta _*$ preserves degrees of $0$-cycles it suffices to compute the degree of $\Delta ^![\Delta ]$. The class $\Delta ^![\Delta ]$ is given by capping $[\Delta ]$ with the top Chern class of the normal sheaf of $\Delta \subset X \times X$ (Chow Homology, Lemma 42.54.5). Since the conormal sheaf of $\Delta $ is $\Omega _{X/k}$ (Morphisms, Lemma 29.32.7) we see that the normal sheaf is equal to the tangent sheaf $\mathcal{T}_ X = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/k}, \mathcal{O}_ X)$ as desired.

Equality on the left. By Lemma 45.7.4 we have

\begin{align*} \deg ([\Delta ] \cdot [\Delta ]) & = \int _{X \times X} \gamma ([\Delta ]) \cup \gamma ([\Delta ]) \\ & = \int _{X \times X} \Delta _*1 \cup \gamma ([\Delta ]) \\ & = \int _{X \times X} \Delta _*(\Delta ^*\gamma ([\Delta ])) \\ & = \int _ X \Delta ^*\gamma ([\Delta ]) \end{align*}

Write $\gamma ([\Delta ]) = \sum e_{i, j} \otimes e'_{2d - i , j}$ as in Lemma 45.7.7. Recalling that $\Delta ^*$ is given by cup product we obtain

\[ \int _ X \sum \nolimits _{i, j} e_{i, j} \cup e'_{2d - i, j} = \sum \nolimits _{i, j} \int _ X e_{i, j} \cup e'_{2d - i, j} = \sum \nolimits _{i, j} (-1)^ i = \sum (-1)^ i\beta _ i \]

as desired. $\square$


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Your email address will not be published.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 0FH0. Beware of the difference between the letter 'O' and the digit '0'.