Class groups of localities of rings of invariants of reductive algebraic groups
Class groups of localities of rings of invariants of reductive algebraic groups
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还原代数群不变量环的局部性类群
DOI：
10.1007/bf01162589
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发表时间：
1983
影响因子：
0.8
通讯作者：
H. Nakajima
中科院分区：
文献类型：
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作者：
H. Nakajima
令 R 为域 k 上的仿射域，令 ~ 为在 k 上定义的仿射代数群，对 R 有 k 有理作用。然后 ~ 3 的 k 有理点群 G (k) 作为 k 代数自同构作用于 R，我们用 Re 表示 k-子代数，由在 ~(k) 作用下的 R 的所有不变元素组成。假设 R 存在一个 G (k) 稳定的最大理想 m，此时 R 是正规的。 (Re(k))~,nRe(k)的约数类群的研究是代数不变量理论中一个重要而古老的课题，该群是在一定的假设下进行计算的。现在我们假设 (R e~ k)) he, k) 是解析正态的（例如，如果 R~ k~] st] 在 k 上有限生成，则满足该假设）。那么就可以考虑R k) 的mc~ R~ k)-进补R e4k) 的除数类群以及该群是否同构于Cl ((Re (k)) m~ Re (k)) 的问题。一般来说，CI ((ReXk)),,~ R~ k)) 并不总是同构 CI (~)，即使 (13 是有限的并且 R 的 m-adic 完成是阶乘的 (参见(3.8))。另一方面，Fossum-Griffith [-21 和 Almkvist-Fossum [1] 表明，当 R 是特征域 k 上的多项式环时，在某些附加条件下，Re (k) 是阶乘的p> 0 且 ~ 3 是循环 p 群，最近 Griffith 在[4]中证明了以下显着结果：
Let R be an affine domain over a field k and let~ be an affine algebraic group defined over k acting k-rationally on R. Then the group G (k) of krational points of~ 3 acts on R as k-algebra automorphisms and we denote by R e~ k) the k-subalgebra consisting of all invariant elements of R under this action of~(k). Suppose that there is a G (k)-stable maximal ideal m of R at which R is normal. The study of the divisor class group of (Re (k))~, nRe (k) is an important and old subject of algebraic invariant theory and this group has been computed with certain hypothesis. Now we assume that (R e~ k)) he, k) is analytically normal (for example this assumption is satisfied, if R~ k~] st] nitely generated over k). Then one can consider the divisor class group of the mc~ R~ k)-adic completion R e4k) of R k) and the question whether the above group is isomorphic to Cl ((Re (k)) m~ Re (k)). In general, CI ((ReXk)),,~ R~ k)) is not always isomorphic CI (~), even if (13 is finite and the m-adic completion of R is factorial (cf.(3.8)). On the other handFossum-Griffith [-21 and Almkvist-Fossum [1] showed that R e (k) is factorial under some additional conditions when R is a polynomial ring over a field k of characteristic p> 0 and~ 3 is a cyclic p-group. Recently Griffith proved in [4] the following remarkable result: