非半单群环上的有限模
Finite modules over non-semisimple group rings
摘要：
设$G$是$n$阶的阿贝尔群，$R$是允许一个同态${\Bbb Z}[ζ_{n}]\ra R$的交换环，其中$ζ_{n}$是（复）本原$n$ -次单位根.给定一个有限的$R[G\e]$ -模$M$，我们推导出一个公式，该公式将$M$的阶与$M$的各个同型分量$M^{\eχ}$的阶的乘积联系起来，其中$χ$的范围在$G$的$R$值特征标群上.然后，我们给出的条件下，$M$的顺序是完全等于产品的顺序的$M^χ$。为了得到这些条件，我们建立在E.Aljadeff的工作，并获得，作为我们的考虑的副产品，一个新的标准上同调平凡，提高了著名的标准T. Nakayama。我们也给出了应用到交换簇和交换域的类群，特别是得到了一些新的类数公式。我们的结果也有“非半单”岩泽理论的应用，但我们不开发这些在这里。一般来说，本文的结果可以用来加强各种已知的结果，涉及有限的$R[G\e]$ -模，其假设包括（一个等价形式）以下假设："$G$的顺序是可逆的$R$“。
英文摘要：
Let $G$ be an abelian group of order $n$ and let $R$ be a commutative ring which admits a homomorphism ${\Bbb Z}[ζ_{n}]\ra R$, where $ζ_{n}$ is a (complex) primitive $n$-th root of unity. Given a finite $R[G\e]$-module $M$, we derive a formula relating the order of $M$ to the product of the orders of the various isotypic components $M^{\eχ}$ of $M$, where $χ$ ranges over the group of $R$-valued characters of $G$. We then give conditions under which the order of $M$ is exactly equal to the product of the orders of the $M^χ$. To derive these conditions, we build on work of E.Aljadeff and obtain, as a by-product of our considerations, a new criterion for cohomological triviality which improves the well-known criterion of T.Nakayama. We also give applications to abelian varieties and to class groups of abelian fields, obtaining in particular some new class number formulas. Our results also have applications to "non-semisimple" Iwasawa theory, but we do not develop these here. In general, the results of this paper can be used to strengthen a variety of known results involving finite $R[G\e]$-modules whose hypotheses include (an equivalent form of) the following assumption: ``the order of $G$ is invertible in $R$".}
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