Theorem 64.14.3 (Lefschetz Trace Formula). Let X be a projective curve over a finite field k, \Lambda a finite ring and K \in D_{ctf}(X, \Lambda ). Then the global and local Lefschetz numbers of K are equal, i.e.,
64.14.3.1
\begin{equation} \label{trace-equation-trace-formula} \text{Tr}(\pi ^*_ X |_{R\Gamma (X_{\bar k}, K)}) = \sum \nolimits _{x\in X(k)} \text{Tr}(\pi _ X |_{K_{\bar x}}) \end{equation}
in \Lambda ^\natural .
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