54.63 Locally constant sheaves

This section is the analogue of Modules on Sites, Section 18.42 for the étale site.

Definition 54.63.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a sheaf of sets on $X_{\acute{e}tale}$.

1. Let $E$ be a set. We say $\mathcal{F}$ is the constant sheaf with value $E$ if $\mathcal{F}$ is the sheafification of the presheaf $U \mapsto E$. Notation: $\underline{E}_ X$ or $\underline{E}$.

2. We say $\mathcal{F}$ is a constant sheaf if it is isomorphic to a sheaf as in (1).

3. We say $\mathcal{F}$ is locally constant if there exists a covering $\{ U_ i \to X\}$ such that $\mathcal{F}|_{U_ i}$ is a constant sheaf.

4. We say that $\mathcal{F}$ is finite locally constant if it is locally constant and the values are finite sets.

Let $\mathcal{F}$ be a sheaf of abelian groups on $X_{\acute{e}tale}$.

1. Let $A$ be an abelian group. We say $\mathcal{F}$ is the constant sheaf with value $A$ if $\mathcal{F}$ is the sheafification of the presheaf $U \mapsto A$. Notation: $\underline{A}_ X$ or $\underline{A}$.

2. We say $\mathcal{F}$ is a constant sheaf if it is isomorphic as an abelian sheaf to a sheaf as in (1).

3. We say $\mathcal{F}$ is locally constant if there exists a covering $\{ U_ i \to X\}$ such that $\mathcal{F}|_{U_ i}$ is a constant sheaf.

4. We say that $\mathcal{F}$ is finite locally constant if it is locally constant and the values are finite abelian groups.

Let $\Lambda$ be a ring. Let $\mathcal{F}$ be a sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$.

1. Let $M$ be a $\Lambda$-module. We say $\mathcal{F}$ is the constant sheaf with value $M$ if $\mathcal{F}$ is the sheafification of the presheaf $U \mapsto M$. Notation: $\underline{M}_ X$ or $\underline{M}$.

2. We say $\mathcal{F}$ is a constant sheaf if it is isomorphic as a sheaf of $\Lambda$-modules to a sheaf as in (1).

3. We say $\mathcal{F}$ is locally constant if there exists a covering $\{ U_ i \to X\}$ such that $\mathcal{F}|_{U_ i}$ is a constant sheaf.

Lemma 54.63.2. Let $f : X \to Y$ be a morphism of schemes. If $\mathcal{G}$ is a locally constant sheaf of sets, abelian groups, or $\Lambda$-modules on $Y_{\acute{e}tale}$, the same is true for $f^{-1}\mathcal{G}$ on $X_{\acute{e}tale}$.

Proof. Holds for any morphism of topoi, see Modules on Sites, Lemma 18.42.2. $\square$

Lemma 54.63.3. Let $f : X \to Y$ be a finite étale morphism of schemes. If $\mathcal{F}$ is a (finite) locally constant sheaf of sets, (finite) locally constant sheaf of abelian groups, or (finite type) locally constant sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$, the same is true for $f_*\mathcal{F}$ on $Y_{\acute{e}tale}$.

Proof. The construction of $f_*$ commutes with étale localization. A finite étale morphism is locally isomorphic to a disjoint union of isomorphisms, see Étale Morphisms, Lemma 40.18.3. Thus the lemma says that if $\mathcal{F}_ i$, $i = 1, \ldots , n$ are (finite) locally constant sheaves of sets, then $\prod _{i = 1, \ldots , n} \mathcal{F}_ i$ is too. This is clear. Similarly for sheaves of abelian groups and modules. $\square$

Lemma 54.63.4. Let $X$ be a scheme and $\mathcal{F}$ a sheaf of sets on $X_{\acute{e}tale}$. Then the following are equivalent

1. $\mathcal{F}$ is finite locally constant, and

2. $\mathcal{F} = h_ U$ for some finite étale morphism $U \to X$.

Proof. A finite étale morphism is locally isomorphic to a disjoint union of isomorphisms, see Étale Morphisms, Lemma 40.18.3. Thus (2) implies (1). Conversely, if $\mathcal{F}$ is finite locally constant, then there exists an étale covering $\{ X_ i \to X\}$ such that $\mathcal{F}|_{X_ i}$ is representable by $U_ i \to X_ i$ finite étale. Arguing exactly as in the proof of Descent, Lemma 34.36.1 we obtain a descent datum for schemes $(U_ i, \varphi _{ij})$ relative to $\{ X_ i \to X\}$ (details omitted). This descent datum is effective for example by Descent, Lemma 34.34.1 and the resulting morphism of schemes $U \to X$ is finite étale by Descent, Lemmas 34.20.23 and 34.20.29. $\square$

Lemma 54.63.5. Let $X$ be a scheme.

1. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of sets on $X_{\acute{e}tale}$. If $\mathcal{F}$ is finite locally constant, there exists an étale covering $\{ U_ i \to X\}$ such that $\varphi |_{U_ i}$ is the map of constant sheaves associated to a map of sets.

2. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of abelian groups on $X_{\acute{e}tale}$. If $\mathcal{F}$ is finite locally constant, there exists an étale covering $\{ U_ i \to X\}$ such that $\varphi |_{U_ i}$ is the map of constant abelian sheaves associated to a map of abelian groups.

3. Let $\Lambda$ be a ring. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of $\Lambda$-modules on $X_{\acute{e}tale}$. If $\mathcal{F}$ is of finite type, then there exists an étale covering $\{ U_ i \to X\}$ such that $\varphi |_{U_ i}$ is the map of constant sheaves of $\Lambda$-modules associated to a map of $\Lambda$-modules.

Proof. This holds on any site, see Modules on Sites, Lemma 18.42.3. $\square$

Lemma 54.63.6. Let $X$ be a scheme.

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

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

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

Proof. This holds on any site, see Modules on Sites, Lemma 18.42.5. $\square$

Lemma 54.63.7. Let $X$ be a scheme. Let $\Lambda$ be a ring. The tensor product of two locally constant sheaves of $\Lambda$-modules on $X_{\acute{e}tale}$ is a locally constant sheaf of $\Lambda$-modules.

Proof. This holds on any site, see Modules on Sites, Lemma 18.42.6. $\square$

Lemma 54.63.8. Let $X$ be a connected scheme. Let $\Lambda$ be a ring and let $\mathcal{F}$ be a locally constant sheaf of $\Lambda$-modules. Then there exists a $\Lambda$-module $M$ and an étale covering $\{ U_ i \to X\}$ such that $\mathcal{F}|_{U_ i} \cong \underline{M}|_{U_ i}$.

Proof. Choose an étale covering $\{ U_ i \to X\}$ such that $\mathcal{F}|_{U_ i}$ is constant, say $\mathcal{F}|_{U_ i} \cong \underline{M_ i}_{U_ i}$. Observe that $U_ i \times _ X U_ j$ is empty if $M_ i$ is not isomorphic to $M_ j$. For each $\Lambda$-module $M$ let $I_ M = \{ i \in I \mid M_ i \cong M\}$. As étale morphisms are open we see that $U_ M = \bigcup _{i \in I_ M} \mathop{\mathrm{Im}}(U_ i \to X)$ is an open subset of $X$. Then $X = \coprod U_ M$ is a disjoint open covering of $X$. As $X$ is connected only one $U_ M$ is nonempty and the lemma follows. $\square$

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