The Stacks project

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

is surjective because the maps

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

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

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

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

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

(Modules on Sites, Lemma 18.43.4) since $\mathcal{F}/I^ n\mathcal{F}$ is (étale) 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

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. $\square$

Proof. By Lemma 61.19.9 we have $\mathcal{F}/I^ n\mathcal{F} = \epsilon ^{-1}\mathcal{G}_ n$ for some locally constant sheaf $\mathcal{G}_ n$ of $\Lambda /I^ n$-modules. By Étale Cohomology, Lemma 59.64.8 there exists a finite $\Lambda /I^ n$-module $M_ n$ such that $\mathcal{G}_ n$ is locally isomorphic to $\underline{M_ n}$. Choose a covering $\{ W_ t \to X\} _{t \in T}$ with each $W_ t$ affine and weakly contractible. Then $\mathcal{F}|_{W_ t}$ satisfies the assumptions of Lemma 61.28.3 and hence $\mathcal{F}|_{W_ t} \cong \underline{N_ t}^\wedge $ for some finite $\Lambda ^\wedge $-module $N_ t$. Note that $N_ t/I^ nN_ t \cong M_ n$ for all $t$ and $n$. Hence $N_ t \cong N_{t'}$ for all $t, t' \in T$, see More on Algebra, Lemma 15.100.5. This proves that $\mathcal{F}$ is adic lisse. $\square$

Proof. This is a consequence of Lemmas 61.28.2 and 61.28.4, the fact that a Noetherian scheme is locally connected (Topology, Lemma 5.9.6), and the definitions. $\square$

Proof. Assume that $\mathcal{F}/I\mathcal{F}$ is zero. Let $I = (f_1, \ldots , f_ r)$. Let $i < r$ be the largest integer such that $\mathcal{G} = \mathcal{F}/(f_1, \ldots , f_ i)\mathcal{F}$ is nonzero. If $i$ does not exist, then $\mathcal{F} = 0$ which is what we want to show. Then $\mathcal{G}$ is derived complete as a cokernel of a map between derived complete modules, see Proposition 61.21.1. By our choice of $i$ we have that $f_{i + 1} : \mathcal{G} \to \mathcal{G}$ is surjective. Hence

is nonzero, contradicting the derived completeness of $\mathcal{G}$. $\square$

Proof. Since $X$ is weakly contractible, there exists a finite disjoint open covering $X = \coprod U_ i$ such that $\mathcal{F}/I\mathcal{F}|_{U_ i}$ is isomorphic to the constant sheaf associated to a finite $\Lambda /I$-module $M_ i$. Choose finitely many generators $m_{ij}$ of $M_ i$. We can find sections $s_{ij} \in \mathcal{F}(X)$ restricting to $m_{ij}$ viewed as a section of $\mathcal{F}/I\mathcal{F}$ over $U_ i$. Let $t$ be the total number of $s_{ij}$. Then we obtain a map

which is surjective modulo $I$ by construction. By Lemma 61.20.1 the derived completion of $\underline{\Lambda }^{\oplus t}$ is the sheaf $(\underline{\Lambda }^\wedge )^{\oplus t}$. Since $\mathcal{F}$ is derived complete we see that $\alpha $ factors through a map

Then $\mathcal{Q} = \mathop{\mathrm{Coker}}(\alpha ^\wedge )$ is a derived complete sheaf of $\Lambda $-modules by Proposition 61.21.1. By construction $\mathcal{Q}/I\mathcal{Q} = 0$. It follows from Lemma 61.28.6 that $\mathcal{Q} = 0$ which is what we wanted to show. $\square$