Definition 54.70.1. 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 $i$.

Comment #73 by Keenan Kidwell on

Should the $X_i$ be locally closed subschemes of $X$ instead of just locally closed subsets?

There are also:

• 2 comment(s) on Section 54.70: Constructible sheaves

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).