FullText URL JofAlgebra_459_76.pdf
Author Hashimoto, Mitsuyasu|
Abstract 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.
Keywords Invariant theory Class group Picard group Krull ring
Note © 2016. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
Published Date 2016-08-01
Publication Title Journal of Algebra
Volume volume459
Start Page 76
End Page 108
ISSN 0021-8693
NCID AA00692420
Content Type Journal Article
language 英語
OAI-PMH Set 岡山大学
Copyright Holders http://creativecommons.org/licenses/by-nc-nd/4.0/
File Version author
DOI 10.1016/j.jalgebra.2016.02.025
Web of Sience KeyUT 000377319700004
Related Url isVersionOf https://doi.org/10.1016/j.jalgebra.2016.02.025