Equivariant class group I. Finite generation of the Picard and the class groups of an invariant subring
Equivariant class group I. Finite generation of the Picard and the class groups of an invariant subring
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等变类群 I. Picard 的有限生成和不变子环的类群
DOI：
10.1016/j.jalgebra.2016.02.025
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发表时间：
2016
影响因子：
0.9
通讯作者：
Mitsuyasu Hashimoto
中科院分区：
文献类型：
--
作者：
Mitsuyasu Hashimoto
本文的目的是定义具有平面群格式作用的局部Krull格式（即局部Krull域的素谱格式）的等变类群，研究其基本性质，并将其应用于证明不变子群的类群的有限生成。特别地，我们证明了以下几点。设k为一个域，G为有限型光滑k群格式，X为拟紧拟分离局部Krull G格式。设有一个有限型k-格式Z和一个显性k-态射Z→X，设φ: X→Y是一个G不变态射，使得oy→（φ O X） G是一个同构。那么Y是局部的Krull。如果Cl (X)是有限生成的，则Cl （G， X）和Cl (Y)也是有限生成的，其中Cl （G， X）是等变类群。事实上，Cl (Y)是Cl （G， X）的子商。对于连通群方案对仿射方案的作用，有Magid和Waterhouse的类似结果，但我们的结果也适用于不连通g。证明依赖于（等变）Picard群的类似结果。
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.