Theorem 64.20.2 (03V0): Adic sheaves

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Part 3: Topics in Scheme Theory
Chapter 64: The Trace Formula
Section 64.20: Cohomological interpretation
Theorem 64.20.2: Adic sheaves (cite)

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Theorem 64.20.2 (Adic sheaves). Let $X$ be a scheme of finite type over a finite field $k$ , and $\mathcal{F}$ a $\mathbf{Q}_\ell$ -sheaf on $X$ . Then

$L(X, \mathcal{F}) = \prod \nolimits _ i \det (1 - \pi _ X^*T |_{H_ c^ i(X_{\bar k} , \mathcal{F})})^{(-1)^{i + 1}} \in \mathbf{Q}_\ell [[T]].$

Proof. This is sketched below. $\square$

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