Chevalley's theorem on constructible sets

Builds on Morphism of schemes Noetherian scheme Morphism of finite type Zariski topology

Statement

Let $Y$ be a Noetherian scheme and let $f \colon X \to Y$ be a morphism of finite type. A subset of a Noetherian scheme is constructible if it is a finite union of sets $U \cap Z$ . Here each $U$ is open and each $Z$ is closed in the Zariski topology. Then $f(C)$ is a constructible subset of $Y$ for every constructible subset $C$ of $X$ . In particular, the image $f(X)$ is constructible.

Remarks

The proof reduces to the case where $Y = \operatorname{Spec} A$ is affine and integral and $X$ is affine. The key step shows that if $f(X)$ is dense in $Y$ , then $f(X)$ contains a nonempty open subset of $Y$ . This follows from generic freeness or from Noether normalization over the fraction field of $A$ . Noetherian induction on the closed complement then finishes the proof. For example, over a field $k$ , the morphism $\mathbb{A}^2_k \to \mathbb{A}^2_k$ given by $(x, y) \mapsto (x, xy)$ has image $\set{u \ne 0} \cup \set{(0, 0)}$ . This image is constructible but neither open nor closed, so constructible sets are the right class for images. Without the finite type hypothesis the image can fail to be constructible. For instance, $\operatorname{Spec} \Q \to \operatorname{Spec} \Z$ has image the generic point alone, which is not constructible.

Going deeper

For a student

Over an algebraically closed field, the theorem is the geometric form of quantifier elimination for algebraically closed fields. It is used to show that properties of the fibres of a morphism hold on constructible sets of the base. Together with generic flatness, it gives the usual stratifications of the base of a family into finitely many pieces over which the family is well behaved. A general version holds for morphisms of finite presentation between arbitrary schemes, with a suitable definition of constructible sets.

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Morphism of finite type concept
Morphism of schemes concept
Noetherian scheme concept
Zariski topology concept
Chevalley's theorem on constructible sets statement

Tag	`TGN01264C4`
URI	`m2://TGN01264C4`
Kind	statement, theorem
Writing	W0 stub
MSC	14A15
Status	This copy of the site has no build data. Run `m2 status TGN01264C4`.
About	Morphism of finite type
Uses	Morphism of schemes, Noetherian scheme, Morphism of finite type, Zariski topology