Karoubi-Villamayor and Quillen functors for triangular categories (Q1076131)

We compute the Bass K-functors \(K_ i\), the K-functors \(k_ i\) of Karoubi-Villamayor and of Quillen, and the Quillen K-functors \(G_ i\) of some trivial ring extensions. Theorem: Let R,S be associative rings, M an R-S-bimodule, N an S-R-bimodule, and \(f: R\times S\to \left( \begin{matrix} R\quad M\\ N\quad S\end{matrix} \right)\) the inclusion. Then (a) \(k_ if\) is an isomorphism for all \(i\in {\mathbb{Z}}\); (b) \(K_ if\) is an isomorphism for all \(i\leq 1\), but already \(K_ 2f\) need not be an isomorphism, in general; (c) if \(N=0\), then \(K_ if\) is an isomorphism for all \(i\in {\mathbb{Z}}\); (d) if \(\left( \begin{matrix} R\quad M\\ N\quad S\end{matrix} \right)\) is coherent from the right, \(N=0\), and for each finitely presented right R-module P there is an \(n\in {\mathbb{Z}}\) such that \(Tor^ R_ i(P,M)=0\) for \(i\geq n\), then \(G_ if\) is an isomorphism for all \(i\geq 0\).