0
$\begingroup$

Let G, X and Y are algebraic schemes over k.(k:field) Assume that G is affine, and that the action is proper. Then f:X -> Y is affine. This is the Proposition0.7 in 'GIT(mumford & Fogarty)' I don't understand the part of the proof of this Prop.

g: G×Y -> X is a proper morphism. P_2: G×Y -> Y is the second projection and affine morphism. and f·g=p_2. The author says "by Chevalley's Theorem(EGA 2, Theorem 6.7.1), f is affine". I can't draw it. Please, help me...

(*Chevalley's Thm: X: affine scheme, Y: noetherian pre-scheme, f: x -> Y is a finite surjective morphism Then Y is also affine.)

$\endgroup$
2
  • $\begingroup$ Which action of $G$ on what? $\endgroup$ Nov 11, 2011 at 10:30
  • $\begingroup$ This is not well-posed in my opinion. Do you mean that $G$ is to be an affine group scheme and that furthermore $G$ acts on $X$ and that $Y$ is the quotient of $X$ modulo $G$? Maybe you should first edit the question in order to make it precise. $\endgroup$ Nov 25, 2011 at 11:36

1 Answer 1

1
$\begingroup$

$g$ is also affine (EGA II, 1.6.1 (v)), hence finite (EGA III, 4.4.2).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.