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

TL;DR

This paper introduces the equivariant class group for schemes with group actions, proves its properties, and demonstrates finite generation of class groups of invariant subrings, extending previous results to disconnected group schemes.

Contribution

It defines the equivariant class group for locally Krull schemes with group actions and proves finite generation of class groups of invariant subrings, including disconnected group schemes.

Findings

The equivariant class group is finitely generated under certain conditions.

The class group of an invariant subring is finitely generated and related to the equivariant class group.

Results extend known theorems to disconnected group schemes.

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 dominating $k$-morphism $Z\rightarrow X$. Let $\varphi:X\rightarrow Y$ be a $G$-invariant morphism such that $\mathcal O_Y\rightarrow (\varphi_*\mathcal O_X)^G$ 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.