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 has its origins in algebraic geometry, where in étale cohomology constructible sheaves are defined in a similar way . 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 on schemes are étale sheaves unless otherwise noted. A sheaf is called constructible if can be written as a finite union of locally closed subschemes such that for each subscheme of the covering, the sheaf is a finite locally constant sheaf. In particular, this means for each subscheme appearing in the finite covering, there is an étale covering such that for all étale subschemes in the cover of , the sheaf 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 to is constant as well, where is the reduction of the scheme . It then follows that a representable étale sheaf 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). 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.