Lemma 59.83.7. In Situation 59.83.1 assume $X$ is smooth. Let $j : U \to X$ an open immersion. Let $\ell$ be a prime number. Let $\mathcal{F} = j_!\underline{\mathbf{Z}/\ell \mathbf{Z}}$. Then statements (1) – (8) hold for $\mathcal{F}$.

Proof. Since $X$ is smooth, it is a disjoint union of smooth curves and hence we may assume $X$ is a curve (i.e., irreducible). Then either $U = \emptyset$ and there is nothing to prove or $U \subset X$ is dense. In this case the lemma follows from Lemmas 59.83.2 and 59.83.6. $\square$

## Comments (2)

Comment #3204 by Alexander Schmidt on

replace "spectrum of an irreducible curve" by "spectrum of an algebraically closed field"

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