ID | 54561 |
フルテキストURL | |
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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.
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キーワード | Invariant theory
Class group
Picard group
Krull ring
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備考 | © 2016. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
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発行日 | 2016-08-01
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出版物タイトル |
Journal of Algebra
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巻 | 459巻
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出版者 | ACADEMIC PRESS INC ELSEVIER SCIENCE
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開始ページ | 76
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終了ページ | 108
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ISSN | 0021-8693
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NCID | AA00692420
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資料タイプ |
学術雑誌論文
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言語 |
英語
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OAI-PMH Set |
岡山大学
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著作権者 | http://creativecommons.org/licenses/by-nc-nd/4.0/
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論文のバージョン | author
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DOI | |
Web of Science KeyUT | |
関連URL | isVersionOf https://doi.org/10.1016/j.jalgebra.2016.02.025
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