The Stacks project

64.2 The trace formula

A typical course in étale cohomology would normally state and prove the proper and smooth base change theorems, purity and Poincaré duality. All of these can be found in [Arcata, SGA4.5]. Instead, we are going to study the trace formula for the frobenius, following the account of Deligne in [Rapport, SGA4.5]. We will only look at dimension 1, but using proper base change this is enough for the general case. Since all the cohomology groups considered will be étale, we drop the subscript $_{\acute{e}tale}$. Let us now describe the formula we are after. Let $X$ be a finite type scheme of dimension 1 over a finite field $k$, $\ell $ a prime number and $\mathcal{F}$ a constructible, flat $\mathbf{Z}/\ell ^ n\mathbf{Z}$ sheaf. Then

64.2.0.1
\begin{equation} \label{trace-equation-trace-formula-initial} \sum \nolimits _{x \in X(k)} \text{Tr}(\text{Frob} | \mathcal{F}_{\bar x}) = \sum \nolimits _{i = 0}^2 (-1)^ i \text{Tr}(\pi _ X^* | H^ i_ c(X \otimes _ k \bar k, \mathcal{F})) \end{equation}

as elements of $\mathbf{Z}/\ell ^ n\mathbf{Z}$. As we will see, this formulation is slightly wrong as stated. Let us nevertheless describe the symbols that occur therein.

Comments (0)

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Your email address will not be published.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 03SJ. Beware of the difference between the letter 'O' and the digit '0'.



View Section 64.2 as pdf