The Stacks project

Lemma 61.28.2. Let $X$ be a Noetherian scheme. Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal. Let $\mathcal{F}$ be a constructible $\Lambda $-sheaf on $X_{pro\text{-}\acute{e}tale}$. Then there exists a finite partition $X = \coprod X_ i$ by locally closed subschemes such that the restriction $\mathcal{F}|_{X_ i}$ is lisse.

Proof. Let $R = \bigoplus I^ n/I^{n + 1}$. Observe that $R$ is a Noetherian ring. Since each of the sheaves $\mathcal{F}/I^ n\mathcal{F}$ is a constructible sheaf of $\Lambda /I^ n\Lambda $-modules also $I^ n\mathcal{F}/I^{n + 1}\mathcal{F}$ is a constructible sheaf of $\Lambda /I$-modules and hence the pullback of a constructible sheaf $\mathcal{G}_ n$ on $X_{\acute{e}tale}$ by Lemma 61.27.2. Set $\mathcal{G} = \bigoplus \mathcal{G}_ n$. This is a sheaf of $R$-modules on $X_{\acute{e}tale}$ and the map

\[ \mathcal{G}_0 \otimes _{\Lambda /I} \underline{R} \longrightarrow \mathcal{G} \]

is surjective because the maps

\[ \mathcal{F}/I\mathcal{F} \otimes \underline{I^ n/I^{n + 1}} \to I^ n\mathcal{F}/I^{n + 1}\mathcal{F} \]

are surjective. Hence $\mathcal{G}$ is a constructible sheaf of $R$-modules by Étale Cohomology, Proposition 59.74.1. Choose a partition $X = \coprod X_ i$ such that $\mathcal{G}|_{X_ i}$ is a locally constant sheaf of $R$-modules of finite type (Étale Cohomology, Lemma 59.71.2). We claim this is a partition as in the lemma. Namely, replacing $X$ by $X_ i$ we may assume $\mathcal{G}$ is locally constant. It follows that each of the sheaves $I^ n\mathcal{F}/I^{n + 1}\mathcal{F}$ is locally constant. Using the short exact sequences

\[ 0 \to I^ n\mathcal{F}/I^{n + 1}\mathcal{F} \to \mathcal{F}/I^{n + 1}\mathcal{F} \to \mathcal{F}/I^ n\mathcal{F} \to 0 \]

induction and Modules on Sites, Lemma 18.43.5 the lemma follows. $\square$


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