In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space X, such that X is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf.
It has its origins in algebraic geometry, where in étale cohomology constructible sheaves are defined in a similar way (Artin, Grothendieck & Verdier 1972, Exposé IX § 2).
For the derived category of constructible sheaves, see a section in ℓ-adic sheaf.
The finiteness theorem in étale cohomology states that the higher direct images of a constructible sheaf are constructible.
Here we use the definition of constructible étale sheaves from the book by Freitag and Kiehl referenced below.
In what follows in this subsection, all sheaves
are étale sheaves unless otherwise noted.
is called constructible if
can be written as a finite union of locally closed subschemes
of the covering, the sheaf
is a finite locally constant sheaf.
appearing in the finite covering, there is an étale covering
such that for all étale subschemes in the cover of
is constant and represented by a finite set.
This definition allows us to derive, from Noetherian induction and the fact that an étale sheaf is constant if and only if its restriction from
It then follows that a representable étale sheaf
Of particular interest to the theory of constructible étale sheaves is the case in which one works with constructible étale sheaves of Abelian groups.
The remarkable result is that constructible étale sheaves of Abelian groups are precisely the Noetherian objects in the category of all torsion étale sheaves (cf.
Most examples of constructible sheaves come from intersection cohomology sheaves or from the derived pushforward of a local system on a family of topological spaces parameterized by a base space.
One nice set of examples of constructible sheaves come from the derived pushforward (with or without compact support) of a local system on
For example, we can set the monodromy operators to be where the stalks of our local system
we get a constructible sheaf where the stalks at the points
compute the cohomology of the local systems restricted to a neighborhood of them in
For example, consider the family of degenerating elliptic curves over
this family of curves degenerates into a nodal curve.
then and where the stalks of the local system