
Equivariant class group. I. Finite generation of the Picard and the class groups of an invariant subringEquivariant class group. I. Finite generation of the Picard and the class groups of an invariant subringAA00692420 
"/Hashimoto, Mitsuyasu/"Hashimoto, Mitsuyasu
459pp.76

108 , 20160801 , ACADEMIC PRESS INC ELSEVIER SCIENCE
ISSN:00218693
NCID:AA00692420
Description
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 kgroup scheme of finite type, and X a quasicompact quasiseparated locally Krull Gscheme. Assume that there is a kscheme Z of finite type and a dominant k morphism Z→XZ→X. Let φ:X→Yφ:X→Y be a G invariant morphism such that OY→(φ⁎OX)GOY→(φ⁎OX)G is an isomorphism. Then Y is locally Krull. If, moreover, Cl(X)Cl(X) is finitely generated, then Cl(G,X)Cl(G,X) and Cl(Y)Cl(Y) are also finitely generated, where Cl(G,X)Cl(G,X) is the equivariant class group.In fact, Cl(Y)Cl(Y) is a subquotient of Cl(G,X)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.