Definition 59.71.1. Let $X$ be a scheme.

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 $i$.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 $i$.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 $i$.

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