In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a constructible sheaf is a
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s over some
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
''X'', such that ''X'' is the union of a finite number of
locally closed subset In topology, a branch of mathematics, a subset E of a topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More sp ...
s on each of which the sheaf is a locally constant sheaf. It has its origins in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, where in
étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjecture ...
constructible sheaves are defined in a similar way . For the derived category of constructible sheaves, see a section in
ℓ-adic sheaf In algebraic geometry, an ℓ-adic sheaf on a Noetherian scheme ''X'' is an inverse system consisting of \mathbb/\ell^n-modules F_n in the étale topology and F_ \to F_n inducing F_ \otimes_ \mathbb/\ell^n \overset\to F_n..
Bhatt–Scholze's pro-à ...
.
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 In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them ov ...
sheaves or from the derived pushforward of a
local system
In mathematics, a local system (or a system of local coefficients) on a topological space ''X'' is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group ''A'', and general sheaf cohomology ...
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
*
References
*
*
*{{Citation
, last1 = Freitag
, first1 = Eberhard
, last2 = Kiehl
, first2 = Reinhardt
, title = Etale Cohomology and the Weil Conjecture
, series = Ergebnisse der Mathematik und ihrer Grenzgebiete (3)
, volume = 13
, publisher = Springer-Verlag
, location = Berlin
, year = 1988
, isbn = 3-540-12175-7
, doi = 10.1007/978-3-662-02541-3
, mr = 926276
Algebraic geometry
Sheaf theory