symmetric monoidal (∞,1)-category of spectra
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 $Y$.
The relation to the elimination of quantifiers for the theory of algebraically closed fields is discussed in
Last revised on May 9, 2016 at 10:00:38. See the history of this page for a list of all contributions to it.