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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
\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.

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