\(K_1\) of twisted rings of polynomials (Q1284151)

Let \(R\) be a ring with identity element 1 and let \(\alpha:R\to R\) be a ring endomorphism that preserves the identity. Let \(R_\alpha[t]\) be the twisted (skew) polynomial ring, in which \(t\cdot r=\alpha(r)\cdot t\) for \(r\) in \(R\). The author shows that there is a direct sum decomposition \[ K_1\bigl(R_\alpha[t]\bigr)\cong K_1(R)\oplus\widetilde{\text{Nil}}(R;\alpha) \] where \(\widetilde{\text{Nil}}(R;\alpha)\) is the Grothendieck group of the category with objects \((F,\varphi)\), \(F\) a finitely generated projective \(R\)-module, \(\varphi\) an \(\alpha\)-linear nilpotent endomorphism of \(F\). A morphism \(f:(F,\varphi)\to(F',\varphi')\) is an \(R\)-linear map on modules with \(\varphi'f=f\varphi\). This result generalizes the results of Bass \((\alpha=id)\) and Farrell-Hsiang \((\alpha\) an automorphism). The argument is to show that the new result follows from the Farrell-Hsiang result, by replacing \(R\) by the direct limit \(R'\) of the system \(R@>\alpha>>R@>\alpha>>R\cdots\), on which \(\alpha\) induces an automorphism.