Chevalley’s theorem

Chevalley’s theorem
Name	Chevalley’s theorem
Field	Algebraic geometry, Algebraic groups, Commutative algebra
Statement	Image of a morphism of schemes of finite type over a field is a constructible set
Introduced	1950s
Author	Claude Chevalley

Chevalley’s theorem is a fundamental result in algebraic geometry asserting that the image of a morphism of finite type between schemes (or of a morphism of algebraic varieties) is a constructible set; equivalently, for morphisms of finite presentation the image is a finite union of locally closed subsets. The theorem links structural results about morphisms with topological properties of algebraic sets and underpins major developments in the theory of algebraic groups, scheme theory, and number theory. It has influenced work in areas connected with the names Grothendieck, Serre, Weil, Langlands, and others.

Statement

The classical form states that for a morphism f: X → Y of schemes of finite type over a field K (or more generally for a morphism of finite presentation between Noetherian schemes), the set f(X) is a constructible subset of Y. In the setting of varieties over an algebraically closed field (as treated by Zariski, Hilbert, and Noether), this says that the image is a finite union of locally closed subvarieties. Important formulations appear in the foundational texts of Grothendieck and Dieudonné and are used in works by Serre, Weil, and Tate.

Historical context and motivation

Chevalley formulated the result in the milieu of mid-20th century algebraic geometry developing out of work by Hilbert, Noether, and Zariski and contemporaneous advances by Grothendieck and Serre. The theorem addressed questions arising in the study of algebraic groups (as in Chevalley’s own program tying linear algebraic groups to arithmetic topics related to Dirichlet and Dedekind), and in elimination theory connected with Hilbert’s Nullstellensatz and the work of Emmy Noether. Influences include the ideas of Hilbert on invariants, the foundations laid in Zariski’s school, and later categorical perspectives introduced by Grothendieck in SGA and EGA, where Chevalley’s statement was integrated into the general machinery of schemes and morphisms.

Proofs and variations

Original proofs combined classical elimination theory and dimension arguments reminiscent of work by Hilbert and Krull, while modern proofs use scheme-theoretic techniques from Grothendieck’s Éléments de géométrie algébrique (EGA) and spectral methods familiar to Serre and Tate. Variations include versions for morphisms of finite presentation, for maps between Noetherian topological spaces, and for constructibility in étale topology as developed by Deligne and Artin. Chevalley’s result is often proved by reduction to the case of affine morphisms and then applying Noether normalization, Zariski’s main theorem, and dimension-theoretic lemmas appearing in the literature of Krull, Noether, and Kronecker.

Applications and consequences

Chevalley’s theorem underlies many structural results: it is used in the theory of algebraic groups as in Chevalley’s decompositions and in the structure theory influencing Weyl, Cartan, and Langlands. It is crucial in the proof of semicontinuity theorems appearing in Grothendieck’s work and in the formulation of the Chebotarev density theorem in arithmetic geometry contexts studied by Frobenius and Chebotarev. The theorem is applied in moduli problems treated by Mumford and Deligne, in elimination theory in the tradition of Hilbert and Ritt, and in the proof of properties of images of morphisms used by Serre, Tate, and Artin. It also supports comparison results involving Hodge theory as in Griffiths and periods studied by Hodge and Deligne.

Examples and counterexamples

Typical examples include projection maps from affine space considered by Hilbert and Noether where images are constructible; maps arising from polynomial maps studied in the work of Bézout and Sylvester; and quotient maps by group actions as in instances treated by Weyl and Emmy Noether where orbits produce locally closed strata. Counterexamples to stronger statements appear in contexts such as infinite type morphisms or morphisms between non-Noetherian schemes connected to examples investigated by Nagata and Zariski: images may fail to be constructible when hypotheses (finite type, finite presentation, Noetherian) are dropped. Other instructive cases arise in the study of algebraic stacks developed by Deligne and Mumford where naive image statements require refinement.

Generalizations and related results

Generalizations include EGA’s formulation for morphisms of finite presentation, constructibility theorems in étale and fppf topologies by Artin and Grothendieck, and extensions to algebraic spaces and algebraic stacks as in work by Deligne, Mumford, Laumon, and Moret-Bailly. Related results are Zariski’s main theorem, generic flatness theorems of Grothendieck, Nagata’s compactification theorem, and semicontinuity theorems used by Kodaira and Spencer in deformation theory. The theorem sits alongside classical elimination and invariant theory from Hilbert, Noether, and Weyl, and it remains a basic tool in modern arithmetic geometry pursued by Langlands, Fontaine, and Faltings.

Category:Algebraic geometry