Algebraic \(K\)-theory of parameterized endomorphisms
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. Neeman} and \textit{A. Ranicki}, ``Noncommutative localization in algebraic \(K\)- theory. I'' [Geom. Topol. 8, 1385-1425 (2004; Zbl 1083.18007)] to show that one obtains isomorphisms \[ \tilde{K}_n(\text{End}(R,M)) = \tilde{K}_{n+1}(\sigma^{-1}TM) \] between the reduced \(K\)-groups.
scientific article; zbMATH DE number 5125367