The Stacks project

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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