start-ver=1.4
cd-journal=joma
no-vol=459
cd-vols=
no-issue=
article-no=
start-page=76
end-page=108
dt-received=
dt-revised=
dt-accepted=
dt-pub-year=2016
dt-pub=20160801
dt-online=
en-article=
kn-article=
en-subject=
kn-subject=
en-title=
kn-title=Equivariant class group. I. Finite generation of the Picard and the class groups of an invariant subring
en-subtitle=
kn-subtitle=
en-abstract=
kn-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→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.
en-copyright=
kn-copyright=
en-aut-name=HashimotoMitsuyasu
en-aut-sei=Hashimoto
en-aut-mei=Mitsuyasu
kn-aut-name=橋本光靖
kn-aut-sei=橋本
kn-aut-mei=光靖
aut-affil-num=1
affil-num=1
en-affil=Okayama University
kn-affil=岡山大学大学院自然科学研究科
en-keyword=Invariant theory
kn-keyword=Invariant theory
en-keyword=Class group
kn-keyword=Class group
en-keyword=Picard group
kn-keyword=Picard group
en-keyword=Krull ring
kn-keyword=Krull ring
END