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.

Examples

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 $\mathbb {P} ^{1}$

One nice set of examples of constructible sheaves come from the derived pushforward (with or without compact support) of a local system on $U=\mathbb {P} ^{1}-\{0,1,\infty \}$ . Since any loop around $\infty$ is homotopic to a loop around $0,1$ we only have to describe the monodromy around $0$ and $1$ . For example, we can set the monodromy operators to be

{\begin{aligned}T_{0}={\begin{bmatrix}1&k\\0&1\end{bmatrix}}&T_{1}={\begin{bmatrix}1&l\\0&1\end{bmatrix}}\end{aligned}}

where the stalks of our local system ${\mathcal {L}}$ are isomorphic to $\mathbb {Q} ^{\oplus 2}$ . Then, if we take the derived pushforward $\mathbf {R} j_{*}$ or $\mathbf {R} j_{!}$ of ${\mathcal {L}}$ for $j:U\to \mathbb {P} ^{1}$ we get a constructible sheaf where the stalks at the points $0,1,\infty$ compute the cohomology of the local systems restricted to a neighborhood of them in $U$ .

Weierstrass Family of Elliptic Curves

For example, consider the family of degenerating elliptic curves

y^{2}-x(x-1)(x-t)

over $\mathbb {C}$ . At $t=0,1$ this family of curves degenerates into a nodal curve. If we denote this family by $\pi :X\to \mathbb {C}$ then

\mathbf {R} ^{0}\pi _{*}({\underline {\mathbb {Q} }}_{X})\cong \mathbf {R} ^{2}\pi _{*}({\underline {\mathbb {Q} }}_{X})\cong {\underline {\mathbb {Q} }}_{\mathbb {C} }

and

\mathbf {R} ^{1}\pi _{*}({\underline {\mathbb {Q} }}_{X})\cong {\mathcal {L}}_{\mathbb {C} -\{0,1\}}\oplus {\underline {\mathbb {Q} }}_{\{0,1\}}

where the stalks of the local system ${\mathcal {L}}_{\mathbb {C} -\{0,1\}}$ are isomorphic to $\mathbb {Q} ^{2}$ . This local monodromy around of this local system around $0,1$ can be computed using the Picard-Lefschetz formula

References

Seminar Notes

Gunningham, Sam; Hughes, Richard, Topics in D-Modules (PDF)

References

Deligne, Pierre, ed. (1977), Séminaire de Géométrie Algébrique du Bois Marie — Cohomologie étale (SGA 4.5), 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

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