In this section we define the notion of a constructible $\Lambda$-sheaf as well as some variants.

Definition 61.28.1. Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. Let $X$ be a scheme. Let $\mathcal{F}$ be a sheaf of $\Lambda$-modules on $X_{pro\text{-}\acute{e}tale}$.

1. We say $\mathcal{F}$ is a constructible $\Lambda$-sheaf if $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}$ and each $\mathcal{F}/I^ n\mathcal{F}$ is a constructible sheaf of $\Lambda /I^ n$-modules.

2. If $\mathcal{F}$ is a constructible $\Lambda$-sheaf, then we say $\mathcal{F}$ is lisse if each $\mathcal{F}/I^ n\mathcal{F}$ is locally constant.

3. We say $\mathcal{F}$ is adic lisse1 if there exists a $I$-adically complete $\Lambda$-module $M$ with $M/IM$ finite such that $\mathcal{F}$ is locally isomorphic to

$\underline{M}^\wedge = \mathop{\mathrm{lim}}\nolimits \underline{M/I^ nM}.$
4. We say $\mathcal{F}$ is adic constructible2 if for every affine open $U \subset X$ there exists a decomposition $U = \coprod U_ i$ into constructible locally closed subschemes such that $\mathcal{F}|_{U_ i}$ is adic lisse.

The definition of a constructible $\Lambda$-sheaf is equivalent to the one in [Exposé VI, Definition 1.1.1, SGA5] when $\Lambda = \mathbf{Z}_\ell$ and $I = (\ell )$. It is clear that we have the implications

$\xymatrix{ \text{lisse adic} \ar@{=>}[r] \ar@{=>}[d] & \text{adic constructible} \ar@{=>}[d] \\ \text{lisse constructible }\Lambda \text{-sheaf} \ar@{=>}[r] & \text{constructible }\Lambda \text{-sheaf} }$

The vertical arrows can be inverted in some cases (see Lemmas 61.28.2 and 61.28.5). In general neither the category of adic constructible sheaves nor the category of adic constructible sheaves is closed under kernels and cokernels.

Namely, let $X$ be an affine scheme whose underlying topological space $|X|$ is homeomorphic to $\Lambda = \mathbf{Z}_\ell$, see Example 61.6.3. Denote $f : |X| \to \mathbf{Z}_\ell = \Lambda$ a homeomorphism. We can think of $f$ as a section of $\underline{\Lambda }^\wedge$ over $X$ and multiplication by $f$ then defines a two term complex

$\underline{\Lambda }^\wedge \xrightarrow {f} \underline{\Lambda }^\wedge$

on $X_{pro\text{-}\acute{e}tale}$. The sheaf $\underline{\Lambda }^\wedge$ is adic lisse. However, the cokernel of the map above, is not adic constructible, as the isomorphism type of the stalks of this cokernel attains infinitely many values: $\mathbf{Z}/\ell ^ n\mathbf{Z}$ and $\mathbf{Z}_\ell$. The cokernel is a constructible $\mathbf{Z}_\ell$-sheaf. However, the kernel is not even a constructible $\mathbf{Z}_\ell$-sheaf as it is zero a non-quasi-compact open but not zero.

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. $\square$

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 (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

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

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. $\square$

Lemma 61.28.4. Let $X$ be a connected scheme. Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. If $\mathcal{F}$ is a lisse constructible $\Lambda$-sheaf on $X_{pro\text{-}\acute{e}tale}$, then $\mathcal{F}$ is adic lisse.

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. $\square$

Lemma 61.28.5. 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 $\mathcal{F}$ is adic constructible.

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. $\square$

It will be useful to identify the constructible $\Lambda$-sheaves inside the category of derived complete sheaves of $\Lambda$-modules. It turns out that the naive analogue of More on Algebra, Lemma 15.94.5 is wrong in this setting. However, here is the analogue of More on Algebra, Lemma 15.91.7.

Lemma 61.28.6. Let $X$ be a scheme. Let $\Lambda$ be a ring and let $I \subset \Lambda$ be a finitely generated ideal. Let $\mathcal{F}$ be a sheaf of $\Lambda$-modules on $X_{pro\text{-}\acute{e}tale}$. If $\mathcal{F}$ is derived complete and $\mathcal{F}/I\mathcal{F} = 0$, then $\mathcal{F} = 0$.

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

$\mathop{\mathrm{lim}}\nolimits (\ldots \to \mathcal{G} \xrightarrow {f_{i + 1}} \mathcal{G} \xrightarrow {f_{i + 1}} \mathcal{G})$

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

Lemma 61.28.7. Let $X$ be a weakly contractible affine scheme. Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. Let $\mathcal{F}$ be a derived complete sheaf of $\Lambda$-modules on $X_{pro\text{-}\acute{e}tale}$ with $\mathcal{F}/I\mathcal{F}$ a locally constant sheaf of $\Lambda /I$-modules of finite type. Then there exists an integer $t$ and a surjective map

$(\underline{\Lambda }^\wedge )^{\oplus t} \to \mathcal{F}$

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

$\alpha : \underline{\Lambda }^{\oplus t} \longrightarrow \mathcal{F}$

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

$\alpha ^\wedge : (\underline{\Lambda }^\wedge )^{\oplus t} \longrightarrow \mathcal{F}$

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. $\square$

[1] This may be nonstandard notation.
[2] This may be nonstandard notation.

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