## 59.74 Constructible sheaves on Noetherian schemes

If $X$ is a Noetherian scheme then any locally closed subset is a constructible locally closed subset (Topology, Lemma 5.16.1). Hence an abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ is constructible if and only if there exists a finite partition $X = \coprod X_ i$ such that $\mathcal{F}|_{X_ i}$ is finite locally constant. (By convention a partition of a topological space has locally closed parts, see Topology, Section 5.28.) In other words, we can omit the adjective “constructible” in Definition 59.71.1. Actually, the category of constructible sheaves on Noetherian schemes has some additional properties which we will catalogue in this section.

Proposition 59.74.1. Let $X$ be a Noetherian scheme. Let $\Lambda$ be a Noetherian ring.

1. Any sub or quotient sheaf of a constructible sheaf of sets is constructible.

2. The category of constructible abelian sheaves on $X_{\acute{e}tale}$ is a (strong) Serre subcategory of $\textit{Ab}(X_{\acute{e}tale})$. In particular, every sub and quotient sheaf of a constructible abelian sheaf on $X_{\acute{e}tale}$ is constructible.

3. The category of constructible sheaves of $\Lambda$-modules on $X_{\acute{e}tale}$ is a (strong) Serre subcategory of $\textit{Mod}(X_{\acute{e}tale}, \Lambda )$. In particular, every submodule and quotient module of a constructible sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$ is constructible.

Proof. Proof of (1). Let $\mathcal{G} \subset \mathcal{F}$ with $\mathcal{F}$ a constructible sheaf of sets on $X_{\acute{e}tale}$. Let $\eta \in X$ be a generic point of an irreducible component of $X$. By Noetherian induction it suffices to find an open neighbourhood $U$ of $\eta$ such that $\mathcal{G}|_ U$ is locally constant. To do this we may replace $X$ by an étale neighbourhood of $\eta$. Hence we may assume $\mathcal{F}$ is constant and $X$ is irreducible.

Say $\mathcal{F} = \underline{S}$ for some finite set $S$. Then $S' = \mathcal{G}_{\overline{\eta }} \subset S$ say $S' = \{ s_1, \ldots , s_ t\}$. Pick an étale neighbourhood $(U, \overline{u})$ of $\overline{\eta }$ and sections $\sigma _1, \ldots , \sigma _ t \in \mathcal{G}(U)$ which map to $s_ i$ in $\mathcal{G}_{\overline{\eta }} \subset S$. Since $\sigma _ i$ maps to an element $s_ i \in S' \subset S = \Gamma (X, \mathcal{F})$ we see that the two pullbacks of $\sigma _ i$ to $U \times _ X U$ are the same as sections of $\mathcal{G}$. By the sheaf condition for $\mathcal{G}$ we find that $\sigma _ i$ comes from a section of $\mathcal{G}$ over the open $\mathop{\mathrm{Im}}(U \to X)$ of $X$. Shrinking $X$ we may assume $\underline{S'} \subset \mathcal{G} \subset \underline{S}$. Then we see that $\underline{S'} = \mathcal{G}$ by Lemma 59.73.12.

Let $\mathcal{F} \to \mathcal{Q}$ be a surjection with $\mathcal{F}$ a constructible sheaf of sets on $X_{\acute{e}tale}$. Then set $\mathcal{G} = \mathcal{F} \times _\mathcal {Q} \mathcal{F}$. By the first part of the proof we see that $\mathcal{G}$ is constructible as a subsheaf of $\mathcal{F} \times \mathcal{F}$. This in turn implies that $\mathcal{Q}$ is constructible, see Lemma 59.71.6.

Proof of (3). we already know that constructible sheaves of modules form a weak Serre subcategory, see Lemma 59.71.6. Thus it suffices to show the statement on submodules.

Let $\mathcal{G} \subset \mathcal{F}$ be a submodule of a constructible sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$. Let $\eta \in X$ be a generic point of an irreducible component of $X$. By Noetherian induction it suffices to find an open neighbourhood $U$ of $\eta$ such that $\mathcal{G}|_ U$ is locally constant. To do this we may replace $X$ by an étale neighbourhood of $\eta$. Hence we may assume $\mathcal{F}$ is constant and $X$ is irreducible.

Say $\mathcal{F} = \underline{M}$ for some finite $\Lambda$-module $M$. Then $M' = \mathcal{G}_{\overline{\eta }} \subset M$. Pick finitely many elements $s_1, \ldots , s_ t$ generating $M'$ as a $\Lambda$-module. (This is possible as $\Lambda$ is Noetherian and $M$ is finite.) Pick an étale neighbourhood $(U, \overline{u})$ of $\overline{\eta }$ and sections $\sigma _1, \ldots , \sigma _ t \in \mathcal{G}(U)$ which map to $s_ i$ in $\mathcal{G}_{\overline{\eta }} \subset M$. Since $\sigma _ i$ maps to an element $s_ i \in M' \subset M = \Gamma (X, \mathcal{F})$ we see that the two pullbacks of $\sigma _ i$ to $U \times _ X U$ are the same as sections of $\mathcal{G}$. By the sheaf condition for $\mathcal{G}$ we find that $\sigma _ i$ comes from a section of $\mathcal{G}$ over the open $\mathop{\mathrm{Im}}(U \to X)$ of $X$. Shrinking $X$ we may assume $\underline{M'} \subset \mathcal{G} \subset \underline{M}$. Then we see that $\underline{M'} = \mathcal{G}$ by Lemma 59.73.12.

Proof of (2). This follows in the usual manner from (3). Details omitted. $\square$

The following lemma tells us that every object of the abelian category of constructible sheaves on $X$ is “Noetherian”, i.e., satisfies a.c.c. for subobjects.

Lemma 59.74.2. Let $X$ be a Noetherian scheme. Let $\Lambda$ be a Noetherian ring. Consider inclusions

$\mathcal{F}_1 \subset \mathcal{F}_2 \subset \mathcal{F}_3 \subset \ldots \subset \mathcal{F}$

in the category of sheaves of sets, abelian groups, or $\Lambda$-modules. If $\mathcal{F}$ is constructible, then for some $n$ we have $\mathcal{F}_ n = \mathcal{F}_{n + 1} = \mathcal{F}_{n + 2} = \ldots$.

Proof. By Proposition 59.74.1 we see that $\mathcal{F}_ i$ and $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ are constructible. Then the lemma follows from Lemma 59.71.8. $\square$

Lemma 59.74.3. Let $X$ be a Noetherian scheme.

1. Let $\mathcal{F}$ be a constructible sheaf of sets on $X_{\acute{e}tale}$. There exist an injective map of sheaves

$\mathcal{F} \longrightarrow \prod \nolimits _{i = 1, \ldots , n} f_{i, *}\underline{E_ i}$

where $f_ i : Y_ i \to X$ is a finite morphism and $E_ i$ is a finite set.

2. Let $\mathcal{F}$ be a constructible abelian sheaf on $X_{\acute{e}tale}$. There exist an injective map of abelian sheaves

$\mathcal{F} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} f_{i, *}\underline{M_ i}$

where $f_ i : Y_ i \to X$ is a finite morphism and $M_ i$ is a finite abelian group.

3. Let $\Lambda$ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$. There exist an injective map of sheaves of modules

$\mathcal{F} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} f_{i, *}\underline{M_ i}$

where $f_ i : Y_ i \to X$ is a finite morphism and $M_ i$ is a finite $\Lambda$-module.

Moreover, we may assume each $Y_ i$ is irreducible, reduced, maps onto an irreducible and reduced closed subscheme $Z_ i \subset X$ such that $Y_ i \to Z_ i$ is finite étale over a nonempty open of $Z_ i$.

Proof. Proof of (1). Because we have the ascending chain condition for subsheaves of $\mathcal{F}$ (Lemma 59.74.2), it suffices to show that for every point $x \in X$ we can find a map $\varphi : \mathcal{F} \to f_*\underline{E}$ where $f : Y \to X$ is finite and $E$ is a finite set such that $\varphi _{\overline{x}} : \mathcal{F}_{\overline{x}} \to (f_*S)_{\overline{x}}$ is injective. (This argument can be avoided by picking a partition of $X$ as in Lemma 59.71.2 and constructing a $Y_ i \to X$ for each irreducible component of each part.) Let $Z \subset X$ be the induced reduced scheme structure (Schemes, Definition 26.12.5) on $\overline{\{ x\} }$. Since $\mathcal{F}$ is constructible, there is a finite separable extension $K/\kappa (x)$ such that $\mathcal{F}|_{\mathop{\mathrm{Spec}}(K)}$ is the constant sheaf with value $E$ for some finite set $E$. Let $Y \to Z$ be the normalization of $Z$ in $\mathop{\mathrm{Spec}}(K)$. By Morphisms, Lemma 29.53.13 we see that $Y$ is a normal integral scheme. As $K/\kappa (x)$ is a finite extension, it is clear that $K$ is the function field of $Y$. Denote $g : \mathop{\mathrm{Spec}}(K) \to Y$ the inclusion. The map $\mathcal{F}|_{\mathop{\mathrm{Spec}}(K)} \to \underline{E}$ is adjoint to a map $\mathcal{F}|_ Y \to g_*\underline{E} = \underline{E}$ (Lemma 59.73.13). This in turn is adjoint to a map $\varphi : \mathcal{F} \to f_*\underline{E}$. Observe that the stalk of $\varphi$ at a geometric point $\overline{x}$ is injective: we may take a lift $\overline{y} \in Y$ of $\overline{x}$ and the commutative diagram

$\xymatrix{ \mathcal{F}_{\overline{x}} \ar@{=}[r] \ar[d] & (\mathcal{F}|_ Y)_{\overline{y}} \ar@{=}[d] \\ (f_*\underline{E})_{\overline{x}} \ar[r] & \underline{E}_{\overline{y}} }$

proves the injectivity. We are not yet done, however, as the morphism $f : Y \to Z$ is integral but in general not finite1.

To fix the problem stated in the last sentence of the previous paragraph, we write $Y = \mathop{\mathrm{lim}}\nolimits _{i \in I} Y_ i$ with $Y_ i$ irreducible, integral, and finite over $Z$. Namely, apply Properties, Lemma 28.22.13 to $f_*\mathcal{O}_ Y$ viewed as a sheaf of $\mathcal{O}_ Z$-algebras and apply the functor $\underline{\mathop{\mathrm{Spec}}}_ Z$. Then $f_*\underline{E} = \mathop{\mathrm{colim}}\nolimits f_{i, *}\underline{E}$ by Lemma 59.51.7. By Lemma 59.73.8 the map $\mathcal{F} \to f_*\underline{E}$ factors through $f_{i, *}\underline{E}$ for some $i$. Since $Y_ i \to Z$ is a finite morphism of integral schemes and since the function field extension induced by this morphism is finite separable, we see that the morphism is finite étale over a nonempty open of $Z$ (use Algebra, Lemma 10.140.9; details omitted). This finishes the proof of (1).

The proofs of (2) and (3) are identical to the proof of (1). $\square$

In the following lemma we use a standard trick to reduce a very general statement to the Noetherian case.

Lemma 59.74.4. Let $X$ be a quasi-compact and quasi-separated scheme.

1. Let $\mathcal{F}$ be a constructible sheaf of sets on $X_{\acute{e}tale}$. There exist an injective map of sheaves

$\mathcal{F} \longrightarrow \prod \nolimits _{i = 1, \ldots , n} f_{i, *}\underline{E_ i}$

where $f_ i : Y_ i \to X$ is a finite and finitely presented morphism and $E_ i$ is a finite set.

2. Let $\mathcal{F}$ be a constructible abelian sheaf on $X_{\acute{e}tale}$. There exist an injective map of abelian sheaves

$\mathcal{F} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} f_{i, *}\underline{M_ i}$

where $f_ i : Y_ i \to X$ is a finite and finitely presented morphism and $M_ i$ is a finite abelian group.

3. Let $\Lambda$ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$. There exist an injective map of sheaves of modules

$\mathcal{F} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} f_{i, *}\underline{M_ i}$

where $f_ i : Y_ i \to X$ is a finite and finitely presented morphism and $M_ i$ is a finite $\Lambda$-module.

Proof. We will reduce this lemma to the Noetherian case by absolute Noetherian approximation. Namely, by Limits, Proposition 32.5.4 we can write $X = \mathop{\mathrm{lim}}\nolimits _{t \in T} X_ t$ with each $X_ t$ of finite type over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ and with affine transition morphisms. By Lemma 59.73.10 the category of constructible sheaves (of sets, abelian groups, or $\Lambda$-modules) on $X_{\acute{e}tale}$ is the colimit of the corresponding categories for $X_ t$. Thus our constructible sheaf $\mathcal{F}$ is the pullback of a similar constructible sheaf $\mathcal{F}_ t$ over $X_ t$ for some $t$. Then we apply the Noetherian case (Lemma 59.74.3) to find an injection

$\mathcal{F}_ t \longrightarrow \prod \nolimits _{i = 1, \ldots , n} f_{i, *}\underline{E_ i} \quad \text{or}\quad \mathcal{F}_ t \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} f_{i, *}\underline{M_ i}$

over $X_ t$ for some finite morphisms $f_ i : Y_ i \to X_ t$. Since $X_ t$ is Noetherian the morphisms $f_ i$ are of finite presentation. Since pullback is exact and since formation of $f_{i, *}$ commutes with base change (Lemma 59.55.3), we conclude. $\square$

Lemma 59.74.5. Let $X$ be a Noetherian scheme. Let $E \subset X$ be a subset closed under specialization.

1. Let $\mathcal{F}$ be a torsion abelian sheaf on $X_{\acute{e}tale}$ whose support is contained in $E$. Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ is a filtered colimit of constructible abelian sheaves $\mathcal{F}_ i$ such that for each $i$ the support of $\mathcal{F}_ i$ is contained in a closed subset contained in $E$.

2. Let $\Lambda$ be a Noetherian ring and $\mathcal{F}$ a sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$ whose support is contained in $E$. Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ is a filtered colimit of constructible sheaves of $\Lambda$-modules $\mathcal{F}_ i$ such that for each $i$ the support of $\mathcal{F}_ i$ is contained in a closed subset contained in $E$.

Proof. Proof of (1). We can write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathcal{F}_ i$ with $\mathcal{F}_ i$ constructible abelian by Lemma 59.73.2. By Proposition 59.74.1 the image $\mathcal{F}'_ i \subset \mathcal{F}$ of the map $\mathcal{F}_ i \to \mathcal{F}$ is constructible. Thus $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}'_ i$ and the support of $\mathcal{F}'_ i$ is contained in $E$. Since the support of $\mathcal{F}'_ i$ is constructible (by our definition of constructible sheaves), we see that its closure is also contained in $E$, see for example Topology, Lemma 5.23.6.

The proof in case (2) is exactly the same and we omit it. $\square$

[1] If $X$ is a Nagata scheme, for example of finite type over a field, then $Y \to Z$ is finite.

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