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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.)

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  • $\begingroup$ 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

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

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