On the homology of filtered and graded rings
On the homology of filtered and graded rings
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关于滤波环和分级环的同源性
DOI：
10.1016/0022-4049(79)90010-0
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发表时间：
1979
影响因子：
0.8
通讯作者：
L. Grunenfelder
中科院分区：
文献类型：
--
作者：
L. Grunenfelder
本文源于试图寻找群 G 的同源性和与 G 的下中心级数相关的（分级）李环 LG 的同源性之间的关系。众所周知，如果 G 是自由的或自由阿贝尔的，则对于所有 n 30，H,(G)= H,,(LG)。对于任意 G，当然不会出现这样的陈述。然而，我们将为有限生成的幂零群提供相当一般的结果。对于交换环 k（特别是，如果 k 是一个域或如果 k= Z），李环 LG@ k 与与群环 kG 的增广过滤相关的分级环 gr kG 密切相关[12]，[2]。因此，在滤波和关联分级环的更一般框架中讨论上述问题是很自然的。主要工具是谱序列，它起源于交换代数 [13，p.11]。二. 171. 对于任意滤波环R、任意滤波左R模块h4和任意滤波右R模块N，存在初始项E’=To的谱序列？ R (gr N, gr M) 在适当的条件下收敛于 TorR (N, M)。在第 1 节中，我们以最适合我们目的的形式描述了下行过滤。第 2 节讨论滤波环和模块背景下的有限性条件、AR 属性、投影覆盖和最小分辨率。这些概念在第 3 节中至关重要，其中给出了上述谱序列的收敛标准。对于径向过滤环，结果如下：
The present paper grew out of an attempt to find a relationship between the homology of a group G and the homology of the (graded) Lie ring LG associated with the lower central series of G. It is well known that H,(G)= H,,(LG) for all n 30 if G is free or free abelian. Such a statement is of course not to be expected for an arbitrary G. However, we shall present quite general results for finitely generated nilpotent groups.For a commutative ring k (in particular, if k is a field or if k= Z) the Lie ring LG@ k is intimately related to the graded ring gr kG associated with the augmentation filtration of the group ring kG [12],[2]. It is therefore natural to discuss the above question in the more general framework of filtered and associated graded rings. The main tool is a spectral sequence which has its origins in commutative algebra [13, p. II. 171. For any filtered ring R, any filtered left R-module h4 and any filtered right R-module N there is a spectral sequence with initial term E’= To? R (gr N, gr M) which under suitable conditions converges to TorR (N, M). In Section 1 we describe descending filtrations in a form which seems most suitable for our purpose. Section 2 deals with finiteness conditions, the AR-property, projective covers and minimal resolutions in the context of filtered rings and modules. These concepts are of crucial importance in Section 3, where convergence criteria for the above spectral sequence are given. For adically filtered rings the result reads as follows: