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

Mitsuyasu Hashimoto

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


Let S be a Noetherian scheme, φ : 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 φ 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 and that the quotient map is universally submersive. Namely, we prove that if G is a flat S-group scheme of finite type acting on X and φ is its universal strict orbit space, then Y is of finite type (S need not be excellent. Geometric fibers of G can be disconnected and non-reduced). Utilizing the technique used there, we also prove that Y is of finite type if φ is flat. The same is true if S is excellent, φ is proper, and Y is Noetherian.

Original languageEnglish
Pages (from-to)807-814
Number of pages8
JournalKyoto Journal of Mathematics
Issue number4
Publication statusPublished - Feb 2004
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)


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