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ID 54561
フルテキストURL
著者
橋本 光靖 岡山大学大学院自然科学研究科
抄録
The purpose of this paper is to define equivariant class group of a locally Krull scheme (that is, a scheme which is locally a prime spectrum of a Krull domain) with an action of a flat group scheme, study its basic properties, and apply it to prove the finite generation of the class group of an invariant subring. In particular, we prove the following. Let k be a field, G a smooth k-group scheme of finite type, and X a quasi-compact quasi-separated locally Krull G-scheme. Assume that there is a k-scheme Z of finite type and a dominant k -morphism Z→XZ→X. Let φ:X→Yφ:X→Y be a G -invariant morphism such that OY→(φ⁎OX)GOY→(φ⁎OX)G is an isomorphism. Then Y is locally Krull. If, moreover, Cl(X)Cl(X) is finitely generated, then Cl(G,X)Cl(G,X) and Cl(Y)Cl(Y) are also finitely generated, where Cl(G,X)Cl(G,X) is the equivariant class group. In fact, Cl(Y)Cl(Y) is a subquotient of Cl(G,X)Cl(G,X). For actions of connected group schemes on affine schemes, there are similar results of Magid and Waterhouse, but our result also holds for disconnected G. The proof depends on a similar result on (equivariant) Picard groups.
キーワード
Invariant theory
Class group
Picard group
Krull ring
備考
© 2016. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
発行日
2016-08-01
出版物タイトル
Journal of Algebra
459巻
出版者
ACADEMIC PRESS INC ELSEVIER SCIENCE
開始ページ
76
終了ページ
108
ISSN
0021-8693
NCID
AA00692420
資料タイプ
学術雑誌論文
言語
English
OAI-PMH Set
岡山大学
著作権者
http://creativecommons.org/licenses/by-nc-nd/4.0/
論文のバージョン
author
DOI
Web of Sience KeyUT
関連URL
isVersionOf https://doi.org/10.1016/j.jalgebra.2016.02.025