The Stacks project

Definition 59.71.1. 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 $i$.

Comments (4)

Comment #73 by Keenan Kidwell on

Should the be locally closed subschemes of instead of just locally closed subsets?

Comment #5405 by Jackson on

For definition (3), should there be a stronger finiteness condition on than just finite type? It seems to contradict definition (2), as a constructible sheaf of -modules may not be a constructible sheaf of abelian groups.

Comment #5635 by on

You are right that a constructible sheaf of -modules is not the same thing as a constructible sheaf of abelian groups. This is a bit inelegant perhaps, but I think this is the correct definition. I would argue that a -module isn't exactly the same thing as an abelian group. It is just true that the categories are (canonically) equivalent. (As a student once pointed out to me when I taught a basic algebra course : these categories are even isomorphic categories. But I still maintain they are not the same.)

There are also:

  • 2 comment(s) on Section 59.71: Constructible sheaves

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