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 A 2A 2\mathbf{A}^2\to\mathbf{A}^2 which takes (x,y)(x,xy)(x,y)\mapsto (x,x y); the image of the affine plane consists of all points (x,y)(x,y) with x0x\neq 0 union (0,0)(0,0). This set is clearly not locally closed, hence not a quasiaffine subvariety of A 2\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:XYf: X\to Y is a regular morphism of varieties and SXS\subset X is a Zariski constructible set. Then f(S)f(S) is also Zariski constructible.

More generally,

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

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


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