Definition 64.14.2 (03TY)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 64: The Trace Formula
Section 64.14: Lefschetz numbers
Definition 64.14.2 (cite)

numberstags

Definition 64.14.2. With $\Lambda , X, k, K$ as in Definition 64.14.1. 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 64.3, 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 $\Lambda ^\natural$ .

Comments (2)

Comment #8300 by Xiaolong Liu on April 08, 2023 at 05:08

I think we should use $\pi_x$ instead of $\pi_X$ here.

Comment #8926 by Stacks project on April 10, 2024 at 17:11

Thanks and fixed here.

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