重新审视法雷尔的零的非有限性

J.-F. Lafont, S. Prassidis, Kun Wang
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引用次数: 3

摘要

研究了环$R$有限阶自同构上的Farrell nil群。我们证明了任何这样的法雷尔零群要么是平凡的,要么是无限生成的(作为一个阿贝尔群)。在第一个结果的基础上,我们证明了在这样的法雷尔零群中出现的任何有限群都具有无限多重性。如果原有限群是直接和,则有限子群的可数无穷和也表现为直接和。利用这个定理,我们推导了有限指数可数法雷尔群的一个结构定理。最后,作为一个应用,我们证明了如果$V$是任意虚循环群,那么相关的Farrell或Waldhausen nil群总是可以表示为有限群的可数无限副本和,只要它们具有有限指数(在维数$0$中总是如此)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Revisiting Farrell’s nonfiniteness of Nil
We study Farrell Nil-groups associated to a finite order automorphism of a ring $R$. We show that any such Farrell Nil-group is either trivial, or infinitely generated (as an abelian group). Building on this first result, we then show that any finite group that occurs in such a Farrell Nil-group occurs with infinite multiplicity. If the original finite group is a direct summand, then the countably infinite sum of the finite subgroup also appears as a direct summand. We use this to deduce a structure theorem for countable Farrell Nil-groups with finite exponent. Finally, as an application, we show that if $V$ is any virtually cyclic group, then the associated Farrell or Waldhausen Nil-groups can always be expressed as a countably infinite sum of copies of a finite group, provided they have finite exponent (which is always the case in dimension $0$).
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