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The Stacks project

59.64 Locally constant sheaves

This section is the analogue of Modules on Sites, Section 18.43 for the étale site.

Definition 59.64.1. 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 59.64.2. 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 18.43.2. \square

Lemma 59.64.3. 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 41.18.3. 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 59.64.4. 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 41.18.3. 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 35.39.1 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 35.37.1 and the resulting morphism of schemes U \to X is finite étale by Descent, Lemmas 35.23.23 and 35.23.29. \square

Lemma 59.64.5. 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 18.43.3. \square

Lemma 59.64.6. 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 18.43.5. \square

Lemma 59.64.7. 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 18.43.6. \square

Lemma 59.64.8. 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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