Revisiting Farrell's nonfiniteness of Nil (Q887837)

For a ring \(R\), Bass Nil-groups \(NK_i(R)\) for (\(i\in \mathbb{Z}\)) is defined as the kernel of the \(K\)-theory map \(K_i(R[t])\rightarrow K_i(R)\) induced by the mapping \(t \mapsto 0\). In this paper, the authors study Farrell twisted Nil-groups \(NK_i(R; \alpha)\) with finite exponent. This group is defined as follows: For an automorphism \(\alpha : R\rightarrow R\), one defines twisted polynomial ring \(R_{\alpha}[t]\). The group \[ NK_i (R; \alpha) = \mathrm{Ker}(\varepsilon* : K_i(R_{\alpha}[t])\rightarrow K_i(R)) \] induced by natural augmentation map \(\varepsilon : R_{\alpha}[t] \rightarrow R\); \(t\mapsto 0\). For \(\alpha\) = identity automorphism, Farrell Nil-groups reduce to the ordinary Bass Nil-groups. \textit{F. T. Farrell} [Proc. Am. Math. Soc. 65, 215--216 (1977; Zbl 0365.18021)] showed that \(NK_i(R)\) with \(i\leq 1\) are either trivial or infinitely generated. Later, using Farrell's idea, this result was extended by \textit{W. van der Kallen} [in: Proc. int. Congr. Math., Helsinki 1978, Vol. 1, 305--310 (1980; Zbl 0426.18005)] and \textit{A. V. Prasolov} [Math. Notes 32, 484--485 (1983); translation from Mat. Zametki 32, 9--12 (1982; Zbl 0512.18004)] for \(NK_i(R)\) with \(i\geq 1\). Then analog result was proved for \(NK_i(R; \alpha)\) with \(i\leq 1\) (when \(\alpha\) has finite order) by \textit{J. Grunewald} [Algebr. Geom. Topol. 7, 1979--1986 (2007; Zbl 1127.19004)] and \textit{R. Ramos} [Bol. Soc. Mat. Mex., III. Ser. 13, No. 1, 55--64 (2007; Zbl 1178.19002)] independently. In this paper, the authors extend the result for \(NK_i(R; \alpha)\) with \(i\geq 1\). Moreover, they show that under the same assumption, for a finite abelian group \(H\), if \(H\) is a subgroup (direct summand) in \(NK_i(R; \alpha)\), then so is \(\underset{\infty}\bigoplus H\) (direct sum of countably many copies of the group \(H\)). As a consequence of the above result they deduce the following structure theorem for certain Farrell Nil-groups: Theorem. Let \(R\) be a countable ring and \(\alpha: R\rightarrow R\) a ring automorphism of finite order, and \(i\in \mathbb{Z}\). If \(NK_i(R; \alpha)\) has finite exponent, then there exists a finite abelian group \(H\) so that \(NK_i(R; \alpha) \cong \underset{\infty}\bigoplus H\). Finally, they deduce an analog structure theorem for the cokernel of so called ``assembly map''.