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 (Deligne 1977, IV.3). A sheaf of abelian groups on a Noetherian scheme is called constructible if the scheme has a finite cover by subschemes on which the sheaf is locally constant constructible (meaning represented by an étale cover). The constructible sheaves form an abelian category.

The finiteness theorem in étale cohomology states that the higher direct images of a constructible sheaf are constructible.

References

• Deligne, Pierre, ed. (1977), Séminaire de Géométrie Algébrique du Bois Marie — Cohomologie étale (SGA 4¹⁄₂), Lecture notes in mathematics (in French) 569 (569), Berlin: Springer-Verlag, doi:10.1007/BFb0091516, ISBN 978-0-387-08066-6
• Dimca, Alexandru (2004), Sheaves in topology, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-20665-1, MR 2050072