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.

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