### Abstract

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 language | English |
---|---|

Pages (from-to) | 807-814 |

Number of pages | 8 |

Journal | Kyoto Journal of Mathematics |

Volume | 43 |

Issue number | 4 |

Publication status | Published - 2004 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Kyoto Journal of Mathematics*,

*43*(4), 807-814.

**"Geometric quotients are algebraic schemes" based on Fogarty's idea.** / Hashimoto, Mitsuyasu.

Research output: Contribution to journal › Article

*Kyoto Journal of Mathematics*, vol. 43, no. 4, pp. 807-814.

}

TY - JOUR

T1 - "Geometric quotients are algebraic schemes" based on Fogarty's idea

AU - Hashimoto, Mitsuyasu

PY - 2004

Y1 - 2004

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=2442614794&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2442614794&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:2442614794

VL - 43

SP - 807

EP - 814

JO - Journal of Mathematics of Kyoto University

JF - Journal of Mathematics of Kyoto University

SN - 0023-608X

IS - 4

ER -