Karoubi-Villamayor and Quillen functors for triangular categories
type: Item
instance of: scholarly article
title: Karoubi-Villamayor and Quillen functors for triangular categories
zbMATH Open document ID: 0593.18008
DOI: 10.1007/BF01086543
author: A. F. Prasolov
published in: Journal of Soviet Mathematics
publication date: 1983-01-01T00:00:00Z
review text: 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 \(i\geq 0\).
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 16E20
zbMATH DE Number: 3953032
zbMATH Keywords: higher algebraic K-theory
zbMATH Keywords: \(K_ i\)
zbMATH Keywords: Karoubi-Villamayor K-functors
zbMATH Keywords: triangular categories
zbMATH Keywords: Quillen K-functors
zbMATH Keywords: trivial ring extensions
MaRDI profile type: Publication
cites work: Higher algebraic K-theory: I
cites work: Trivial extensions of Abelian categories. Homological algebra of trivial extensions of Abelian categories with applications to ring theory
cites work: K i of Upper Triangular Matrix Rings
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full work available at URL: https://doi.org/10.1007/bf01086543
OpenAlex ID: W2071882935
description: scientific article; zbMATH DE number 3953032