The Stacks project

Proof. without further mention. Write $\mathcal{F} = \mathcal{F}_1 \oplus \ldots \oplus \mathcal{F}_ r$ where $\mathcal{F}_ i$ is annihilated by $\ell _ i^{n_ i}$ for some prime $\ell _ i$ and integer $n_ i > 0$. By Lemma 59.84.5 it suffices to prove the theorem for $\mathcal{F}_ i$. Thus we may and do assume that $\ell ^ n$ kills $\mathcal{F}$ for some prime $\ell $ and integer $n > 0$.

Since $\mathcal{F}$ is constructible as a sheaf of $\Lambda $-modules, there is a dense open $U \subset X$ such that $\mathcal{F}|_ U$ is a finite type, locally constant sheaf of $\Lambda $-modules. Since $\dim (X) \leq 1$ we may assume, after shrinking $U$, that $U = U_1 \amalg \ldots \amalg U_ n$ is a disjoint union of irreducible schemes (just remove the closed points which lie in the intersections of $\geq 2$ components of $U$). By Lemma 59.84.3 we reduce to the case $\mathcal{F} = j_!\mathcal{G}$ where $\mathcal{G}$ is a finite type, locally constant sheaf of $\Lambda $-modules on $U$ (and annihilated by $\ell ^ n$).

Since we chose $U = U_1 \amalg \ldots \amalg U_ n$ with $U_ i$ irreducible we have

where $j_ i : U_ i \to X$ is the inclusion morphism. The case of $j_{i!}(\mathcal{G}|_{U_ i})$ is handled in Lemma 59.84.6. $\square$