In
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, constructible sets are a class of subsets 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 specifically, a topological space is a set whose elements are called points ...
that have a relatively "simple" structure.
They are used particularly 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 ...
and related fields. A key result known as ''Chevalley's theorem''
in algebraic geometry shows that the image of a constructible set is constructible for an important class of
mappings
(more specifically
morphisms
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
) of
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
(or more generally
schemes).
In addition, a large number of "local" geometric properties of schemes, morphisms and sheaves are (locally) constructible.
Constructible sets also feature in the definition of various types of
constructible sheaves in algebraic geometry
and
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 ...
.
Definitions
A simple definition, adequate in many situations, is that a constructible set is a finite
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of
locally closed sets. (A set is locally closed if it is the
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
of an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
and
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
.)
However, a modification and another slightly weaker definition are needed to have definitions that behave better with "large" spaces:
Definitions: A subset
of a topological space
is called ''retrocompact'' if
is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
for every compact open subset
. A subset of
is ''constructible'' if it is a ''finite'' union of subsets of the form
where both
and
are open ''and retrocompact'' subsets of
.
A subset
is ''locally constructible'' if there is a
cover
Cover or covers may refer to:
Packaging
* Another name for a lid
* Cover (philately), generic term for envelope or package
* Album cover, the front of the packaging
* Book cover or magazine cover
** Book design
** Back cover copy, part of co ...
of
consisting of open subsets with the property that each
is a constructible subset of
.
Equivalently the constructible subsets of a topological space
are the smallest collection
of subsets of
that (i) contains all open retrocompact subsets and (ii) contains all
complements and finite unions (and hence also finite intersections) of sets in it. In other words, constructible sets are precisely the Boolean algebra generated by retrocompact open subsets.
In a
locally noetherian topological space In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, s ...
, ''all'' subsets are retrocompact, and so for such spaces the simplified definition given first above is equivalent to the more elaborate one. Most of the commonly met schemes in algebraic geometry (including all
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
) are locally Noetherian, but there are important constructions that lead to more general schemes.
In any (not necessarily
noetherian In mathematics, the adjective Noetherian is used to describe Category_theory#Categories.2C_objects.2C_and_morphisms, objects that satisfy an ascending chain condition, ascending or descending chain condition on certain kinds of subobjects, meaning t ...
) topological space, every constructible set contains a
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
open subset of its closure.
Terminology: The definition given here is the one used by the first edition of
EGA and the
Stacks Project The Stacks Project is an open source collaborative mathematics textbook writing project with the aim to cover "algebraic stacks and the algebraic geometry needed to define them". , the book consists of 115 chapters (excluding the license and index ...
. In the second edition of EGA constructible sets (according to the definition above) are called "globally constructible" while the word "constructible" is reserved for what are called locally constructible above.
Chevalley's theorem
A major reason for the importance of constructible sets in algebraic geometry is that the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of a (locally) constructible set is also (locally) constructible for a large class of maps (or "morphisms"). The key result is:
Chevalley's theorem. If
is a
finitely presented morphism of schemes and
is a locally constructible subset, then
is also locally constructible in
.
In particular, the image of an algebraic variety need not be a variety, but is (under the assumptions) always a constructible set. For example, the map
that sends
to
has image the set
, which is not a variety, but is constructible.
Chevalley's theorem in the generality stated above would fail if the simplified definition of constructible sets (without restricting to ''retrocompact'' open sets in the definition) were used.
Constructible properties
A large number of "local" properties of morphisms of schemes and
quasicoherent sheaves on schemes hold true over a locally constructible subset. EGA IV § 9 covers a large number of such properties. Below are some examples (where all references point to EGA IV):
* If
is an finitely presented morphism of schemes and
is a sequence of finitely presented quasi-coherent
-modules, then the set of
for which
is exact is locally constructible. (Proposition (9.4.4))
* If
is an finitely presented morphism of schemes and
is a finitely presented quasi-coherent
-module, then the set of
for which
is locally free is locally constructible. (Proposition (9.4.7))
* If
is an finitely presented morphism of schemes and
is a locally constructible subset, then the set of
for which
is closed (or open) in
is locally constructible. (Corollary (9.5.4))
* Let
be a scheme and
a morphism of
-schemes. Consider the set
of
for which the induced morphism
of fibres over
has some property
. Then
is locally constructible if
is any of the following properties
: surjective, proper, finite, immersion, closed immersion, open immersion, isomorphism. (Proposition (9.6.1))
* Let
be an finitely presented morphism of schemes and consider the set
of
for which the fibre
has a property
. Then
is locally constructible if
is any of the following properties
: geometrically irreducible, geometrically connected, geometrically reduced. (Theorem (9.7.7))
*Let
be an locally finitely presented morphism of schemes and consider the set
of
for which the fibre
has a property
. Then
is locally constructible if
is any of the following properties
: geometrically regular, geometrically normal, geometrically reduced. (Proposition (9.9.4))
One important role that these constructibility results have is that in most cases assuming the morphisms in questions are also
flat
Flat or flats may refer to:
Architecture
* Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries
Arts and entertainment
* Flat (music), a symbol () which denotes a lower pitch
* Flat (soldier), ...
it follows that the properties in question in fact hold in an ''open'' subset. A substantial number of such results is included in EGA IV § 12.
See also
*
Constructible topology In commutative algebra, the constructible topology on the spectrum \operatorname(A) of a commutative ring A is a topology where each closed set is the image of \operatorname (B) in \operatorname(A) for some algebra ''B'' over ''A''. An important ...
*
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 origin ...
Notes
References
* Allouche, Jean Paul.
Note on the constructible sets of a topological space''
*
*
Borel, Armand. ''Linear algebraic groups.''
*
*
*
*
*
External links
* https://stacks.math.columbia.edu/tag/04ZC Topological definition of (local) constructibility
* https://stacks.math.columbia.edu/tag/054H Constructibility properties of morphisms of schemes (incl. Chevalley's theorem)
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Topology
Algebraic geometry