"Geometric quotients are algebraic schemes" based on Fogarty's idea

Details "Geometric quotients are algebraic schemes" based on Fogarty's idea "Geometric quotients are algebraic schemes" based on Fogarty's idea

2003-03-12

arxiv.org/abs/math/0303134v2

J. Math. Kyoto Univ. 43 (4) (2003), 807-814

Mathematics

Commutative Algebra

10 pages, minor error corrections and change of numbering of theorems

Scientific paper

Let S be a Noetherian scheme, f:X->Y a surjective S-morphism of S-schemes, with X of finite type over S. We discuss what makes Y of finite type. First, we prove that if S is excellent, Y is reduced, and f is universally open, then Y is of finite type. We apply this to understand Fogarty's theorem in "Geometric quotients are algebraic schemes, Adv. Math. 48 (1983), 166--171" for the special case that the group scheme G is flat over the Noetherian base scheme S. Namely, we prove that if G is a flat S-group scheme of finite type acting on X and f is its strict orbit space, then Y is of finite type. Utilizing the technique used there, we also prove that Y is of finite type if f is flat. The same is true if S is excellent, f is proper, and Y is Noetherian.

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