The Stacks project

Lemma 45.7.4. Let $H^*$ be a classical Weil cohomology theory (Definition 45.7.3). Let $X$ be a smooth projective variety of dimension $d$. The diagram

\[ \xymatrix{ \mathop{\mathrm{CH}}\nolimits ^ d(X) \ar[r]_-\gamma \ar@{=}[d] & H^{2d}(X) \ar[d]^{\int _ X} \\ \mathop{\mathrm{CH}}\nolimits _0(X) \ar[r]^\deg & F } \]

commutes where $\deg : \mathop{\mathrm{CH}}\nolimits _0(X) \to \mathbf{Z}$ is the degree of zero cycles discussed in Chow Homology, Section 42.41.

Proof. The result holds for $\mathop{\mathrm{Spec}}(k)$ by axiom (C)(d). Let $x : \mathop{\mathrm{Spec}}(k) \to X$ be a closed point of $X$. Then we have $\gamma ([x]) = x_*\gamma ([\mathop{\mathrm{Spec}}(k)])$ in $H^{2d}(X)$ by axiom (C)(b). Hence $\int _ X \gamma ([x]) = 1$ by the definition of $x_*$. $\square$

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Your email address will not be published.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