Constructible sheaf

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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 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 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).

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 . Since any loop around is homotopic to a loop around we only have to describe the monodromy around and . For example, we can set the monodromy operators to be

where the stalks of our local system are isomorphic to . Then, if we take the derived pushforward or of for 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 .

Weierstrass Family of Elliptic Curves[]

For example, consider the family of degenerating elliptic curves

over . At this family of curves degenerates into a nodal curve. If we denote this family by then

and

where the stalks of the local system are isomorphic to . This local monodromy around of this local system around can be computed using the Picard–Lefschetz formula

References[]

Seminar Notes[]

  • Gunningham, Sam; Hughes, Richard, Topics in D-Modules (PDF), archived from the original (PDF) on 2017-09-21

References[]

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