Theorem 64.20.1 (03UZ): Finite Coefficients

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Part 3: Topics in Scheme Theory
Chapter 64: The Trace Formula
Section 64.20: Cohomological interpretation
Theorem 64.20.1: Finite Coefficients (cite)

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Theorem 64.20.1 (Finite Coefficients). Let $X$ be a scheme of finite type over a finite field $k$ . Let $\Lambda$ be a finite ring of order prime to the characteristic of $k$ and $\mathcal{F}$ a constructible flat $\Lambda$ -module on $X_{\acute{e}tale}$ . Then

$L(X, \mathcal{F}) = \det (1 - \pi _ X^*\ T |_{R\Gamma _ c(X_{\bar k}, \mathcal{F})})^{-1} \in \Lambda [[T]].$

Proof. Omitted. $\square$

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