nLab Chevalley's theorem on constructible sets

An image of a variety under a regular map is not necessarily a variety. A standard example is to take the map $\mathbf{A}^2\to\mathbf{A}^2$ which takes $(x,y)\mapsto (x,x y)$; the image of the affine plane consists of all points $(x,y)$ with $x\neq 0$ union $(0,0)$. This set is clearly not locally closed, hence not a quasiaffine subvariety of $\mathbf{A}^2$.

However, the image of a variety under a regular map is always a constructible set (finite union of locally closed sets). More generally,

THEOREM. (Chevalley) If $f: X\to Y$ is a regular morphism of varieties and $S\subset X$ is a Zariski constructible set. Then $f(S)$ is also Zariski constructible.

More generally,

Theorem (EGA IV 1.8.4.) If $f:X\to Y$ is a finitely presented morphism of schemes. Then the image of any constructible subset of $X$ under $f$ is a constructible subset of $Y$.

Related entries

• tame topology

Links

• MathOverflow chevalleys-theorem-on-constructible-sets

The relation to the elimination of quantifiers for the theory of algebraically closed fields is discussed in

• D. Sustretov, blog: model theory IV: algebraically closed fields and compact complex manifolds

Reference

• A. Joyal, Les Théorèmes de Chevalley-Tarski et Remarques sur l’Algèbre Constructive , Cah. Top. Géom. Diff. Cat. XVI (1975) pp.256-258. (Proceedings Colloque Amiens 1975, 6.81 MB)

category: algebraic geometry