1

it turns out i can't do any of the exercises in section 7.4 of ravi vakil :(

exercise 7.4.M. is:

Show that to prove Chevalley’s Theorem (that the image of constructibl sets under finite type morphism of Noetherian schemes is constructible) , it suffices to prove that if $\pi: X \to Y$ is a finite type morphism of Noetherian schemes, the image of π is constructible.

my meager observations: if the image of any such morphism is constructible, then the image of any open set is constructible. if we knew the complement of a set with constructible image has constructible image we would be done. (but this may be stronger than chevalleys theorem and so is possibly untrue.)

any help is appreciated.

Edit: Thanks to those below. At this point in the text there are many concepts (closed subschemes etc) invoked in the proofs (here and in the duplicate post) that have not been developed by Vakil. in general this section seems a bit sketchier than the neighboring ones, so this may be unavoidable.

user638699
  • 61

1 Answers1

1

I never know what is meant by Chevalley's theorem, everyone has its own version. The version I know is

EGA IV 1 théorème 1.8.4 (that I will call main theorem) : if the morphism of schmes $f : X \to Y$ is locally of finite presentation then for if $Z$ is a locally constructible subset of $Y$, the subset $f(Z)$ is locally constructible in $Y$.

The version that you provide is : EGA IV 1 corollaire 1.8.5 (that I will call reduced theorem).

The strategy to prove the main theorem is quite instructive, so let me remind it by quoting EGA (as it is a delight to read it) : you take $y\in Y$ and $V$ an open affine neighborhood of $y$. As the morphism $f$ is quasi-compact and quasi-separated so is its "restriction" $f^{-1} (V) \to V$ which implies that $f^{-1}$ is a quasi-compact and quasi-separated scheme. Through the instructive EGA IV 1 1.8.1 the part $Z \cap f^{-1} (V)$ is constructible, which shows that it suffices to prove the main theorem with $Y$ affine and $Z$ constructible. The scheme $X$ itself is quasi-compact and quasi-separeted so that you can find a morphism of finite presentation $g : X' \to X$ such that $g(X') = Z$. Then as $f \circ g$ is of finite presentation as well one sees that one can suppose that $Z = X$. That is, one has to show that : if $Y$ is an affine scheme and $f : X \to Y$ is a quasi-compact morphism that is locally of finite presentation then $f(X)$ is a constructible subset of $Y$. (This is actually EGA IV 1 lemme 1.8.4.1.) In this case as $X$ is quasi-compact it is finite union of open affines so that we can suppose $Y = \textrm{Spec}(A)$, $X = \textrm{Spec}(B)$ and that $B$ is $A$-algebra of finite presentation. Now $A$ is the inductive limit of its finite type $\mathbf{Z}$-sub-algebras. Then by the technical EGA IV 1 lemme 1.8.4.2 there is such a finite type $\mathbf{Z}$-sub-algebras $A_0$ and an $A_0$-algebra $B_0$ of finite type such that $B$ is isomorphic to $B_0 \otimes_{A_0} A$. Now if $Y_0 := \textrm{Spec}(A_0)$ and $X_0 = \textrm{Spec}(B_0)$ then $X = X_0 \times_{Y_0} Y$ with the projection $X \to Y$ being equal to $f$. If $f_0 : X_0 \to Y_0$ and $g_0 : Y \to Y_0$ are the structural morphisms one sees (thanks to EGA 1, corollaire 3.4.8) that $f(X) = g_0^{-1} \left( f_0^{-1} \left( X_0 \right)\right)$ : it indeed suffice to show that $f_0 (X_0)$ is constructible, i.e. to show the reduced theorem.

Olórin
  • 12,448