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. $\square$


Comments (0)

There are also:

  • 1 comment(s) on Section 61.28: Constructible adic sheaves

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09BV. Beware of the difference between the letter 'O' and the digit '0'.