### 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 O_{Y}→(φ*O_{X})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 language | English |
---|---|

Pages (from-to) | 76-108 |

Number of pages | 33 |

Journal | Journal of Algebra |

Volume | 459 |

DOIs | |

Publication status | Published - Aug 1 2016 |

### Fingerprint

### Keywords

- Class group
- Invariant theory
- Krull ring
- Picard group

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Equivariant class group. I. Finite generation of the Picard and the class groups of an invariant subring.** / Hashimoto, Mitsuyasu.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Equivariant class group. I. Finite generation of the Picard and the class groups of an invariant subring

AU - Hashimoto, Mitsuyasu

PY - 2016/8/1

Y1 - 2016/8/1

N2 - 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.

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.

KW - Class group

KW - Invariant theory

KW - Krull ring

KW - Picard group

UR - http://www.scopus.com/inward/record.url?scp=84964317989&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84964317989&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2016.02.025

DO - 10.1016/j.jalgebra.2016.02.025

M3 - Article

VL - 459

SP - 76

EP - 108

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -