A unified Kummer-Artin-Schreier sequence
A unified Kummer-Artin-Schreier sequence
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统一的 Kummer-Artin-Schreier 序列
DOI：
10.1007/bf01458325
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发表时间：
1987
影响因子：
1.4
通讯作者：
W. Waterhouse
中科院分区：
文献类型：
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作者：
W. Waterhouse
HI(R，Z/PZ)：Kummer正合列I~7Z/PZ~G，，~G，~i，其中p是可逆的且是本原p次单位根(在R中)，以及当p在R中为零时有效的Artin-Schreier序列1~71/pz~G，~Ga1(见[6，pp.126-127])。在本文中，我将证明这两种情况都是定义在Z[(]上的同一类单一序列的特例。所涉及的两个群(除Z/PZ外)具有可用常见的K-理论计算的H1-上同调，因此我们得到了计算所有Z[(]-代数R的H1(R，7Z/PZ)的方法。特别地，当p=2时，这种计算适用于所有环，并且我们恢复了以前用特殊方法(见例如[7])导出的二次代数的描述。精确序列中出现的群方案是同余子群的光滑版本，我相信在我与Boris Weisfeeller对一维光滑连通群方案进行分类[8]的联合工作中首次对这些对象进行了一般描述。
Hi (R, Z/pZ): the Kummer exact sequence I~ 7Z/pZ~ G,,~ G,,~ I, applicable where p is invertible and a primitive p-th root of unity (is in R, and the Artin-Schreier sequence 1~ 71/pZ~ G,~ Ga 1 that is valid when p is zero in R (see eg [6, pp. 126-127]). In this paper I shall show that these are both special cases of a single sequence of the same kind defined over Z [(]. The two groups involved (besides Z/pZ) have H 1-cohomology computable by familiar K-theory, and hence we have a way of computing H 1 (R, 7Z/pZ) for all Z [(]-algebras R. In particular, the computation works for all rings when p= 2, and we recover the description of etale quadratic algebras derived previously by ad hoc methods (see eg [7]). The group schemes occurring in the exact sequence are smoothed versions of congruence subgroups, objects which (I believe) were first described in general in my joint work with Boris Weisfeiler classifying smooth connected group schemes of dimension one [8].