Equivariant class group. I. Finite generation of the Picard and the class groups of an invariant subring

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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→X. Let φ : X→ Y be a G-invariant morphism such that OY→(φ*OX)G is an isomorphism. Then Y is locally Krull. If, moreover, Cl(X) is finitely generated, then Cl(G, X) and Cl(Y) are also finitely generated, where Cl(G, X) is the equivariant class group. In fact, Cl(Y) is a subquotient of 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.

Original languageEnglish
Pages (from-to)76-108
Number of pages33
JournalJournal of Algebra
Volume459
DOIs
Publication statusPublished - Aug 1 2016

Fingerprint

Class Group
Subring
Equivariant
Group Scheme
Invariant
Finite Type
Morphism
Finitely Generated
Krull Domain
Prime Spectrum
Picard Group
K-group
Isomorphism

Keywords

  • Class group
  • Invariant theory
  • Krull ring
  • Picard group

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

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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→X. Let φ : X→ Y be a G-invariant morphism such that OY→(φ*OX)G is an isomorphism. Then Y is locally Krull. If, moreover, Cl(X) is finitely generated, then Cl(G, X) and Cl(Y) are also finitely generated, where Cl(G, X) is the equivariant class group. In fact, Cl(Y) is a subquotient of 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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AB - 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→X. Let φ : X→ Y be a G-invariant morphism such that OY→(φ*OX)G is an isomorphism. Then Y is locally Krull. If, moreover, Cl(X) is finitely generated, then Cl(G, X) and Cl(Y) are also finitely generated, where Cl(G, X) is the equivariant class group. In fact, Cl(Y) is a subquotient of 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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