In topology, constructible sets are a class of subsets of a topological space that have a relatively "simple" structure. They are used particularly in algebraic geometry 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) of algebraic varieties (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.

Definitions

A simple definition, adequate in many situations, is that a constructible set is a finite union of locally closed sets. (A set is locally closed if it is the intersection of an open set and closed set.) 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 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 of consisting of open subsets with the property that each is a constructible subset of . [1][2]

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, all subsets are retrocompact,[3] 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) are locally Noetherian, but there are important constructions that lead to more general schemes.

In any (not necessarily noetherian) topological space, every constructible set contains a dense open subset of its closure.[4]

Terminology: The definition given here is the one used by the first edition of EGA and the Stacks Project. 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. [5]

Chevalley's theorem

A major reason for the importance of constructible sets in algebraic geometry is that the image 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 .[6][7][8]

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.[9]

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[10] 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 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.[11]

See also

Notes

  1. Grothendieck & Dieudonné 1961, Ch. 0III, Définitions (9.1.1), (9.1.2) and (9.1.11), pp. 12-14
  2. "Definition 5.15.1 (tag 005G)". stacks.math.columbia.edu. Retrieved 2022-10-04.
  3. Grothendieck & Dieudonné 1961, Ch. 0III, Sect. (9.1), p. 12
  4. Jinpeng An (2012). "Rigid geometric structures, isometric actions, and algebraic quotients". Geom. Dedicata 157: 153–185.
  5. Grothendieck & Dieudonné 1971, Ch. 0I, Définitions (2.3.1), (2.3.2) and (2.3.10), pp. 55-57
  6. Grothendieck & Dieudonné 1964, Ch. I, Théorème (1.8.4), p. 239.
  7. "Theorem 29.22.3 (Chevalley's Theorem) (tag 054K)". stacks.math.columbia.edu. Retrieved 2022-10-04.
  8. Grothendieck & Dieudonné 1971, Ch. I, Théorème (7.1.4), p. 329.
  9. "Section 109.24 Images of locally closed subsets (tag 0GZL)". stacks.math.columbia.edu. Retrieved 2022-10-04.
  10. Grothendieck & Dieudonné 1966, Ch. IV, § 9 Propriétés constructibles, pp. 54-94.
  11. Grothendieck & Dieudonné 1966, Ch. IV, § 12 Étude des fibres des morphismes plats de présentation finie, pp. 173-187.

References

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.