Annals of K-Theory

Revisiting Farrell's nonfiniteness of Nil

Jean-François Lafont, Stratos Prassidis, and Kun Wang

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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 zero).

Article information

Ann. K-Theory, Volume 1, Number 2 (2016), 209-225.

Received: 23 February 2015
Accepted: 31 March 2015
First available in Project Euclid: 16 November 2017

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Zentralblatt MATH identifier

Primary: 18F25: Algebraic $K$-theory and L-theory [See also 11Exx, 11R70, 11S70, 12- XX, 13D15, 14Cxx, 16E20, 19-XX, 46L80, 57R65, 57R67] 19D35: Negative $K$-theory, NK and Nil 18E10: Exact categories, abelian categories

Nil-groups algebraic $K$-theory Frobenius functors Verschiebung functors


Lafont, Jean-François; Prassidis, Stratos; Wang, Kun. Revisiting Farrell's nonfiniteness of Nil. Ann. K-Theory 1 (2016), no. 2, 209--225. doi:10.2140/akt.2016.1.209.

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