## 58.70 Constructible sheaves

Let $X$ be a scheme. A constructible locally closed subscheme of $X$ is a locally closed subscheme $T \subset X$ such that the underlying topological space of $T$ is a constructible subset of $X$. If $T, T' \subset X$ are locally closed subschemes with the same underlying topological space, then $T_{\acute{e}tale}\cong T'_{\acute{e}tale}$ by the topological invariance of the étale site (Theorem 58.45.2). Thus in the following definition we may assume our locally closed subschemes are reduced.

Definition 58.70.1. Let $X$ be a scheme.

1. A sheaf of sets on $X_{\acute{e}tale}$ is constructible if for every affine open $U \subset X$ there exists a finite decomposition of $U$ into constructible locally closed subschemes $U = \coprod _ i U_ i$ such that $\mathcal{F}|_{U_ i}$ is finite locally constant for all $i$.

2. A sheaf of abelian groups on $X_{\acute{e}tale}$ is constructible if for every affine open $U \subset X$ there exists a finite decomposition of $U$ into constructible locally closed subschemes $U = \coprod _ i U_ i$ such that $\mathcal{F}|_{U_ i}$ is finite locally constant for all $i$.

3. Let $\Lambda$ be a Noetherian ring. A sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$ is constructible if for every affine open $U \subset X$ there exists a finite decomposition of $U$ into constructible locally closed subschemes $U = \coprod _ i U_ i$ such that $\mathcal{F}|_{U_ i}$ is of finite type and locally constant for all $i$.

It seems that this is the accepted definition. An alternative, which lends itself more readily to generalizations beyond the étale site of a scheme, would have been to define constructible sheaves by starting with $h_ U$, $j_{U!}\mathbf{Z}/n\mathbf{Z}$, and $j_{U!}\underline{\Lambda }$ where $U$ runs over all quasi-compact and quasi-separated objects of $X_{\acute{e}tale}$, and then take the smallest full subcategory of $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$, $\textit{Ab}(X_{\acute{e}tale})$, and $\textit{Mod}(X_{\acute{e}tale}, \underline{\Lambda })$ containing these and closed under finite limits and colimits. It follows from Lemma 58.70.6 and Lemmas 58.72.5, 58.72.7, and 58.72.6 that this produces the same category if $X$ is quasi-compact and quasi-separated. In general this does not produce the same category however.

A disjoint union decomposition $U = \coprod U_ i$ of a scheme by locally closed subschemes will be called a partition of $U$ (compare with Topology, Section 5.28).

Lemma 58.70.2. Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F}$ be a sheaf of sets on $X_{\acute{e}tale}$. The following are equivalent

1. $\mathcal{F}$ is constructible,

2. there exists an open covering $X = \bigcup U_ i$ such that $\mathcal{F}|_{U_ i}$ is constructible, and

3. there exists a partition $X = \bigcup X_ i$ by constructible locally closed subschemes such that $\mathcal{F}|_{X_ i}$ is finite locally constant.

A similar statement holds for abelian sheaves and sheaves of $\Lambda$-modules if $\Lambda$ is Noetherian.

Proof. It is clear that (1) implies (2).

Assume (2). For every $x \in X$ we can find an $i$ and an affine open neighbourhood $V_ x \subset U_ i$ of $x$. Hence we can find a finite affine open covering $X = \bigcup V_ j$ such that for each $j$ there exists a finite decomposition $V_ j = \coprod V_{j, k}$ by locally closed constructible subsets such that $\mathcal{F}|_{V_{j, k}}$ is finite locally constant. By Topology, Lemma 5.15.5 each $V_{j, k}$ is constructible as a subset of $X$. By Topology, Lemma 5.28.7 we can find a finite stratification $X = \coprod X_ l$ with constructible locally closed strata such that each $V_{j, k}$ is a union of $X_ l$. Thus (3) holds.

Assume (3) holds. Let $U \subset X$ be an affine open. Then $U \cap X_ i$ is a constructible locally closed subset of $U$ (for example by Properties, Lemma 28.2.1) and $U = \coprod U \cap X_ i$ is a partition of $U$ as in Definition 58.70.1. Thus (1) holds. $\square$

Lemma 58.70.3. Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F}$ be a sheaf of sets, abelian groups, $\Lambda$-modules (with $\Lambda$ Noetherian) on $X_{\acute{e}tale}$. If there exist constructible locally closed subschemes $T_ i \subset X$ such that (a) $X = \bigcup T_ j$ and (b) $\mathcal{F}|_{T_ j}$ is constructible, then $\mathcal{F}$ is constructible.

Proof. First, we can assume the covering is finite as $X$ is quasi-compact in the spectral topology (Topology, Lemma 5.23.2 and Properties, Lemma 28.2.4). Observe that each $T_ i$ is a quasi-compact and quasi-separated scheme in its own right (because it is constructible in $X$; details omitted). Thus we can find a finite partition $T_ i = \coprod T_{i, j}$ into locally closed constructible parts of $T_ i$ such that $\mathcal{F}|_{T_{i, j}}$ is finite locally constant (Lemma 58.70.2). By Topology, Lemma 5.15.12 we see that $T_{i, j}$ is a constructible locally closed subscheme of $X$. Then we can apply Topology, Lemma 5.28.7 to $X = \bigcup T_{i, j}$ to find the desired partition of $X$. $\square$

Lemma 58.70.4. Let $X$ be a scheme. Checking constructibility of a sheaf of sets, abelian groups, $\Lambda$-modules (with $\Lambda$ Noetherian) can be done Zariski locally on $X$.

Proof. The statement means if $X = \bigcup U_ i$ is an open covering such that $\mathcal{F}|_{U_ i}$ is constructible, then $\mathcal{F}$ is constructible. If $U \subset X$ is affine open, then $U = \bigcup U \cap U_ i$ and $\mathcal{F}|_{U \cap U_ i}$ is constructible (it is trivial that the restriction of a constructible sheaf to an open is constructible). It follows from Lemma 58.70.2 that $\mathcal{F}|_ U$ is constructible, i.e., a suitable partition of $U$ exists. $\square$

Lemma 58.70.5. Let $f : X \to Y$ be a morphism of schemes. If $\mathcal{F}$ is a constructible sheaf of sets, abelian groups, or $\Lambda$-modules (with $\Lambda$ Noetherian) on $Y_{\acute{e}tale}$, the same is true for $f^{-1}\mathcal{F}$ on $X_{\acute{e}tale}$.

Proof. By Lemma 58.70.4 this reduces to the case where $X$ and $Y$ are affine. By Lemma 58.70.2 it suffices to find a finite partition of $X$ by constructible locally closed subschemes such that $f^{-1}\mathcal{F}$ is finite locally constant on each of them. To find it we just pull back the partition of $Y$ adapted to $\mathcal{F}$ and use Lemma 58.63.2. $\square$

Lemma 58.70.6. Let $X$ be a scheme.

1. The category of constructible sheaves of sets is closed under finite limits and colimits inside $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$.

2. The category of constructible abelian sheaves is a weak Serre subcategory of $\textit{Ab}(X_{\acute{e}tale})$.

3. Let $\Lambda$ be a Noetherian ring. The category of constructible sheaves of $\Lambda$-modules on $X_{\acute{e}tale}$ is a weak Serre subcategory of $\textit{Mod}(X_{\acute{e}tale}, \Lambda )$.

Proof. We prove (3). We will use the criterion of Homology, Lemma 12.10.3. Suppose that $\varphi : \mathcal{F} \to \mathcal{G}$ is a map of constructible sheaves of $\Lambda$-modules. We have to show that $\mathcal{K} = \mathop{\mathrm{Ker}}(\varphi )$ and $\mathcal{Q} = \mathop{\mathrm{Coker}}(\varphi )$ are constructible. Similarly, suppose that $0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{G} \to 0$ is a short exact sequence of sheaves of $\Lambda$-modules with $\mathcal{F}$, $\mathcal{G}$ constructible. We have to show that $\mathcal{E}$ is constructible. In both cases we can replace $X$ with the members of an affine open covering. Hence we may assume $X$ is affine. The we may further replace $X$ by the members of a finite partition of $X$ by constructible locally closed subschemes on which $\mathcal{F}$ and $\mathcal{G}$ are of finite type and locally constant. Thus we may apply Lemma 58.63.6 to conclude.

The proofs of (1) and (2) are very similar and are omitted. $\square$

Lemma 58.70.7. Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. The tensor product of two constructible sheaves of $\Lambda$-modules on $X_{\acute{e}tale}$ is a constructible sheaf of $\Lambda$-modules.

Proof. The question immediately reduces to the case where $X$ is affine. Since any two partitions of $X$ with constructible locally closed strata have a common refinement of the same type and since pullbacks commute with tensor product we reduce to Lemma 58.63.7. $\square$

Lemma 58.70.8. Let $X$ be a quasi-compact and quasi-separated scheme.

1. Let $\mathcal{F} \to \mathcal{G}$ be a map of constructible sheaves of sets on $X_{\acute{e}tale}$. Then the set of points $x \in X$ where $\mathcal{F}_{\overline{x}} \to \mathcal{G}_{\overline{x}}$ is surjective, resp. injective, resp. is isomorphic to a given map of sets, is constructible in $X$.

2. Let $\mathcal{F}$ be a constructible abelian sheaf on $X_{\acute{e}tale}$. The support of $\mathcal{F}$ is constructible.

3. Let $\Lambda$ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$. The support of $\mathcal{F}$ is constructible.

Proof. Proof of (1). Let $X = \coprod X_ i$ be a partition of $X$ by locally closed constructible subschemes such that both $\mathcal{F}$ and $\mathcal{G}$ are finite locally constant over the parts (use Lemma 58.70.2 for both $\mathcal{F}$ and $\mathcal{G}$ and choose a common refinement). Then apply Lemma 58.63.5 to the restriction of the map to each part.

The proof of (2) and (3) is omitted. $\square$

The following lemma will turn out to be very useful later on. It roughly says that the category of constructible sheaves has a kind of weak “Noetherian” property.

Lemma 58.70.9. Let $X$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathcal{F}_ i$ be a filtered colimit of sheaves of sets, abelian sheaves, or sheaves of modules.

1. If $\mathcal{F}$ and $\mathcal{F}_ i$ are constructible sheaves of sets, then the ind-object $\mathcal{F}_ i$ is essentially constant with value $\mathcal{F}$.

2. If $\mathcal{F}$ and $\mathcal{F}_ i$ are constructible sheaves of abelian groups, then the ind-object $\mathcal{F}_ i$ is essentially constant with value $\mathcal{F}$.

3. Let $\Lambda$ be a Noetherian ring. If $\mathcal{F}$ and $\mathcal{F}_ i$ are constructible sheaves of $\Lambda$-modules, then the ind-object $\mathcal{F}_ i$ is essentially constant with value $\mathcal{F}$.

Proof. Proof of (1). We will use without further mention that finite limits and colimits of constructible sheaves are constructible (Lemma 58.63.6). For each $i$ let $T_ i \subset X$ be the set of points $x \in X$ where $\mathcal{F}_{i, \overline{x}} \to \mathcal{F}_{\overline{x}}$ is not surjective. Because $\mathcal{F}_ i$ and $\mathcal{F}$ are constructible $T_ i$ is a constructible subset of $X$ (Lemma 58.70.8). Since the stalks of $\mathcal{F}$ are finite and since $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathcal{F}_ i$ we see that for all $x \in X$ we have $x \not\in T_ i$ for $i$ large enough. Since $X$ is a spectral space by Properties, Lemma 28.2.4 the constructible topology on $X$ is quasi-compact by Topology, Lemma 5.23.2. Thus $T_ i = \emptyset$ for $i$ large enough. Thus $\mathcal{F}_ i \to \mathcal{F}$ is surjective for $i$ large enough. Assume now that $\mathcal{F}_ i \to \mathcal{F}$ is surjective for all $i$. Choose $i \in I$. For $i' \geq i$ denote $S_{i'} \subset X$ the set of points $x$ such that the number of elements in $\mathop{\mathrm{Im}}(\mathcal{F}_{i, \overline{x}} \to \mathcal{F}_{\overline{x}})$ is equal to the number of elements in $\mathop{\mathrm{Im}}(\mathcal{F}_{i, \overline{x}} \to \mathcal{F}_{i', \overline{x}})$. Because $\mathcal{F}_ i$, $\mathcal{F}_{i'}$ and $\mathcal{F}$ are constructible $S_{i'}$ is a constructible subset of $X$ (details omitted; hint: use Lemma 58.70.8). Since the stalks of $\mathcal{F}_ i$ and $\mathcal{F}$ are finite and since $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \mathcal{F}_{i'}$ we see that for all $x \in X$ we have $x \not\in S_{i'}$ for $i'$ large enough. By the same argument as above we can find a large $i'$ such that $S_{i'} = \emptyset$. Thus $\mathcal{F}_ i \to \mathcal{F}_{i'}$ factors through $\mathcal{F}$ as desired.

Proof of (2). Observe that a constructible abelian sheaf is a constructible sheaf of sets. Thus case (2) follows from (1).

Proof of (3). We will use without further mention that the category of constructible sheaves of $\Lambda$-modules is abelian (Lemma 58.63.6). For each $i$ let $\mathcal{Q}_ i$ be the cokernel of the map $\mathcal{F}_ i \to \mathcal{F}$. The support $T_ i$ of $\mathcal{Q}_ i$ is a constructible subset of $X$ as $\mathcal{Q}_ i$ is constructible (Lemma 58.70.8). Since the stalks of $\mathcal{F}$ are finite $\Lambda$-modules and since $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathcal{F}_ i$ we see that for all $x \in X$ we have $x \not\in T_ i$ for $i$ large enough. Since $X$ is a spectral space by Properties, Lemma 28.2.4 the constructible topology on $X$ is quasi-compact by Topology, Lemma 5.23.2. Thus $T_ i = \emptyset$ for $i$ large enough. This proves the first assertion. For the second, assume now that $\mathcal{F}_ i \to \mathcal{F}$ is surjective for all $i$. Choose $i \in I$. For $i' \geq i$ denote $\mathcal{K}_{i'}$ the image of $\mathop{\mathrm{Ker}}(\mathcal{F}_ i \to \mathcal{F})$ in $\mathcal{F}_{i'}$. The support $S_{i'}$ of $\mathcal{K}_{i'}$ is a constructible subset of $X$ as $\mathcal{K}_{i'}$ is constructible. Since the stalks of $\mathop{\mathrm{Ker}}(\mathcal{F}_ i \to \mathcal{F})$ are finite $\Lambda$-modules and since $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \mathcal{F}_{i'}$ we see that for all $x \in X$ we have $x \not\in S_{i'}$ for $i'$ large enough. By the same argument as above we can find a large $i'$ such that $S_{i'} = \emptyset$. Thus $\mathcal{F}_ i \to \mathcal{F}_{i'}$ factors through $\mathcal{F}$ as desired. $\square$

Comment #1720 by Keenan Kidwell on

In the last sentence of the first paragraph of this tag, the first instance of the word "are" should be "our."

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