### 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 |
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Pages (from-to) | 76-108 |

Number of pages | 33 |

Journal | Journal of Algebra |

Volume | 459 |

DOIs | |

Publication status | Published - Aug 1 2016 |

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### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory