Definition 63.14.2. With $\Lambda , X, k, K$ as in Definition 63.14.1. Since $K\in D_{ctf}(X, \Lambda )$, for any geometric point $\bar x$ of $X$, the complex $K_{\bar x}$ is a perfect complex (in $D_{perf}(\Lambda )$). As we have seen in Section 63.3, the Frobenius $\pi _ X$ acts on $K_{\bar x}$. The local Lefschetz number of $K$ is the sum

$\sum \nolimits _{x\in X(k)} \text{Tr}(\pi _ X |_{K_{\overline{x}}})$

which is again an element of $\Lambda ^\natural$.

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