The Stacks project

Lemma 61.28.3. Let $X$ be a weakly contractible affine scheme. Let $\Lambda $ be a Noetherian ring and $I \subset \Lambda $ be an ideal. Let $\mathcal{F}$ be a sheaf of $\Lambda $-modules on $X_{pro\text{-}\acute{e}tale}$ such that

  1. $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}$,

  2. $\mathcal{F}/I^ n\mathcal{F}$ is a constant sheaf of $\Lambda /I^ n$-modules,

  3. $\mathcal{F}/I\mathcal{F}$ is of finite type.

Then $\mathcal{F} \cong \underline{M}^\wedge $ where $M$ is a finite $\Lambda ^\wedge $-module.

Proof. Pick a $\Lambda /I^ n$-module $M_ n$ such that $\mathcal{F}/I^ n\mathcal{F} \cong \underline{M_ n}$. Since we have the surjections $\mathcal{F}/I^{n + 1}\mathcal{F} \to \mathcal{F}/I^ n\mathcal{F}$ we conclude that there exist surjections $M_{n + 1} \to M_ n$ inducing isomorphisms $M_{n + 1}/I^ nM_{n + 1} \to M_ n$. Fix a choice of such surjections and set $M = \mathop{\mathrm{lim}}\nolimits M_ n$. Then $M$ is an $I$-adically complete $\Lambda $-module with $M/I^ nM = M_ n$, see Algebra, Lemma 10.98.2. Since $M_1$ is a finite type $\Lambda $-module (Modules on Sites, Lemma 18.42.5) we see that $M$ is a finite $\Lambda ^\wedge $-module. Consider the sheaves

\[ \mathcal{I}_ n = \mathit{Isom}(\underline{M_ n}, \mathcal{F}/I^ n\mathcal{F}) \]

on $X_{pro\text{-}\acute{e}tale}$. Modding out by $I^ n$ defines a transition map

\[ \mathcal{I}_{n + 1} \longrightarrow \mathcal{I}_ n \]

By our choice of $M_ n$ the sheaf $\mathcal{I}_ n$ is a torsor under

\[ \mathit{Isom}(\underline{M_ n}, \underline{M_ n}) = \underline{\text{Isom}_\Lambda (M_ n, M_ n)} \]

(Modules on Sites, Lemma 18.43.4) since $\mathcal{F}/I^ n\mathcal{F}$ is (├ętale) locally isomorphic to $\underline{M_ n}$. It follows from More on Algebra, Lemma 15.100.4 that the system of sheaves $(\mathcal{I}_ n)$ is Mittag-Leffler. For each $n$ let $\mathcal{I}'_ n \subset \mathcal{I}_ n$ be the image of $\mathcal{I}_ N \to \mathcal{I}_ n$ for all $N \gg n$. Then

\[ \ldots \to \mathcal{I}'_3 \to \mathcal{I}'_2 \to \mathcal{I}'_1 \to * \]

is a sequence of sheaves of sets on $X_{pro\text{-}\acute{e}tale}$ with surjective transition maps. Since $*(X)$ is a singleton (not empty) and since evaluating at $X$ transforms surjective maps of sheaves of sets into surjections of sets, we can pick $s \in \mathop{\mathrm{lim}}\nolimits \mathcal{I}'_ n(X)$. The sections define isomorphisms $\underline{M}^\wedge \to \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F} = \mathcal{F}$ and the proof is done. $\square$

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