著者 橋本 光靖|
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 http://creativecommons.org/licenses/by-nc-nd/4.0/
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 © 2016 Elsevier Inc.
論文のバージョン author
DOI 10.1016/j.jalgebra.2016.02.025
Web of Sience KeyUT 000377319700004
オフィシャル URL http://www.sciencedirect.com/science/article/pii/S0021869316300138|