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ID 54561
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Author
Hashimoto, Mitsuyasu Okayama University
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
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
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/
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isVersionOf https://doi.org/10.1016/j.jalgebra.2016.02.025