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

Mitsuyasu Hashimoto

Research output: Contribution to journalArticle

3 Citations (Scopus)

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 languageEnglish
Pages (from-to)807-814
Number of pages8
JournalKyoto Journal of Mathematics
Volume43
Issue number4
Publication statusPublished - 2004
Externally publishedYes

Fingerprint

Finite Type
Quotient
Noetherian
Group Scheme
Quotient Map
Orbit Space
Morphism
Fiber
Theorem

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Kyoto Journal of Mathematics, Vol. 43, No. 4, 2004, p. 807-814.

Research output: Contribution to journalArticle

Hashimoto, Mitsuyasu. / "Geometric quotients are algebraic schemes" based on Fogarty's idea. In: Kyoto Journal of Mathematics. 2004 ; Vol. 43, No. 4. pp. 807-814.
@article{fb6fcc8eb09d4080b2b1c555894f5407,
title = "{"}Geometric quotients are algebraic schemes{"} based on Fogarty's idea",
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.",
author = "Mitsuyasu Hashimoto",
year = "2004",
language = "English",
volume = "43",
pages = "807--814",
journal = "Journal of Mathematics of Kyoto University",
issn = "0023-608X",
publisher = "Kyoto University",
number = "4",

}

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 -