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

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 $Z\to X$. Let $\phi :X\to Y$ be a $G$-invariant morphism such that ${\mathcal{O}}_Y\to ({\phi }_{\ast }{\mathcal{O}}_X{)}^G$ 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.

39 pages

References in corpus (1)

• Equivariant total ring of fractions and factoriality of rings generated by semiinvariants

Cited by in corpus (1)

• Equivariant class group. II. Enriched descent theorem