Theorem 63.14.3 (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.,

63.14.3.1
$$\label{trace-equation-trace-formula} \text{Tr}(\pi ^*_ X |_{R\Gamma (X_{\bar k}, K)}) = \sum \nolimits _{x\in X(k)} \text{Tr}(\pi _ X |_{K_{\bar x}})$$

in $\Lambda ^\natural$.

Proof. See discussion below. $\square$

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