Constructible sheaf

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 is a generalization of constructible topology in classical algebraic geometry.
In étale cohomology constructible sheaves are defined in a similar way .
A sheaf of abelian groups on a Noetherian scheme is called constructible if the scheme has a finite cover by locally closed subschemes on which the sheaf is locally constant constructible (meaning represented by an étale cover). 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.
Definition of étale constructible sheaves on a scheme "X".
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 formula_1 on schemes formula_2 are étale sheaves unless otherwise noted.
A sheaf formula_1 is called constructible if formula_2 can be written as a finite union of locally closed subschemes formula_5 such that for each subscheme formula_6 of the covering, the sheaf formula_7 is a finite locally constant sheaf. In particular, this means for each subscheme formula_6 appearing in the finite covering, there is an étale covering formula_9 such that for all étale subschemes in the cover of formula_6, the sheaf formula_11 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 formula_2 to formula_13 is constant as well, where formula_13 is the reduction of the scheme formula_2. It then follows that a representable étale sheaf formula_1 is itself constructible.
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. Proposition I.4.8 of Freitag-Kiehl).
Examples in algebraic topology.
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.
Derived Pushforward on P1.
One nice set of examples of constructible sheaves come from the derived pushforward (with or without compact support) of a local system on formula_17. Since any loop around formula_18 is homotopic to a loop around formula_19 we only have to describe the monodromy around formula_20 and formula_21. For example, we can set the monodromy operators to be
where the stalks of our local system formula_23 are isomorphic to formula_24. Then, if we take the derived pushforward formula_25 or formula_26 of formula_23 for formula_28 we get a constructible sheaf where the stalks at the points formula_29 compute the cohomology of the local systems restricted to a neighborhood of them in formula_30.
Weierstrass Family of Elliptic Curves.
For example, consider the family of degenerating elliptic curves
over formula_32. At formula_33 this family of curves degenerates into a nodal curve. If we denote this family by formula_34 then
and
where the stalks of the local system formula_37 are isomorphic to formula_38. This local monodromy around of this local system around formula_19 can be computed using the Picard–Lefschetz formula