Algebraic \(K\)-theory of parameterized endomorphisms
type: Item
instance of: scholarly article
title: Algebraic \(K\)-theory of parameterized endomorphisms
zbMATH Open document ID: 1115.18005
published in: \(K\)-Theory
publication date: 2007-02-13T00:00:00Z
review text: For a bimodule \(M\) over a ring \(R\) let End\((R,M)\) denote the following category, called the category of parameterized endomorphisms: objects are pairs \((P,f)\) with \(P\) a finitely generated projective right \(R\)-module and \(f:P \rightarrow P \otimes M\) a homomorphism, and morphisms \(\Phi: (P,f) \rightarrow (Q,g)\) are given by maps \(\phi: P \rightarrow Q\) satisfying \(g \circ \phi = (\phi \otimes id) \circ f.\) Assume now that \(R\) is a commutative semi-simple ring. Let \(TM = \oplus_{i \geq 0}M^{\otimes i}\) denote the tensor algebra on \(M\). Any object \((P,f) \in \text{End}(R,M)\) induces a map \(id - f: P\otimes TM \rightarrow P\otimes TM\), and \((P,f)\) is nilpotent if and only \(id-f\) is an isomorphism. Let \(Nil\) denote the subcategory of End\((R,M)\) of all nilpotent objects. The author deduces from results of \textit{M. Schlichting} [cf. ``Delooping the \(K\)-theory of exact categories'', Topology 43, 1089--1103 (2004; Zbl 1059.18007)]) that there is a homotopy fibration of \(K\)-theory spaces \[ K(Nil) \rightarrow K(\text{End}(R,M)) \rightarrow K(\mathcal H), \] where \(\mathcal H\) is obtained from End\((R,M)\) by formally inverting weak isomorphisms. It is shown that \(\mathcal H\) is equivalent to the full subcategory \(H(TM,E)\) of the category of right \(TM\)-modules, whose objects are cokernels of the maps \(id-f: P \otimes TM \rightarrow P\otimes TM\) with \((P,f) \in \text{End}(R,M)\). As a consequence one obtains a long exact sequence of algebraic \(K\)-theory groups \[ \cdots \rightarrow K_{i+1}(H(TM,E)) \rightarrow K_i(Nil) \rightarrow K_i(\text{End}(R,M)) \rightarrow K_i(H(TM,E)) \rightarrow \cdots \] In the special case that \(R\) is a field the author relates the \(K\)-theory of \(H(TM,E))\) to the \(K\)-theory of the noncommutative localization \(\sigma^{-1}TM\) of \(TM\) with respect to the collection \(\sigma\) of \(TM\)-maps \(id-f: P \otimes TM \rightarrow P\otimes TM\), and uses the localization sequence of \textit{A…
reviewed by: Manfred Kolster
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 5125367
zbMATH Keywords: algebraic \(K\)-theory
zbMATH Keywords: noncommutative localization
zbMATH Keywords: tensor algebra
author: Stanislaw Betley
MaRDI profile type: Publication
full work available at URL: s10977-006-7107-x
OpenAlex ID: W1976211485
cites work: Stable \(K\)-theory and topological Hochschild homology
cites work: The K-theory of endomorphisms
cites work: Noncommutative localisation in algebraic \(K\)-theory. I
cites work: On the cohomology and K-theory of the general linear groups over a finite field
cites work: Delooping the \(K\)-theory of exact categories
cites work: Algebraic K-theory of generalized free products. I
DOI: 10.1007/S10977-006-7107-X
description: scientific article; zbMATH DE number 5125367