Celebratio Mathematica

A. Neeman and A. Ranicki: Noncommutative localization and chain complexes, I: Algebraic \( K \)- and \( L \)-theory. Preprint, September 2001. ArXiv math/0109118 techreport

The noncommutative (Cohn) localization \( \sigma^{-1}R \) of a ring \( R \) is defined for any collection \( \sigma \) of morphisms of f.g. projective left \( R \)-modules. We exhibit \( \sigma^{-1}R \) as the endomorphism ring of \( R \) in an appropriate triangulated category. We use this expression to prove that if \( \sigma^{-1}R \) is “stably flat over R” (meaning that \[ \mathrm{Tor}^R_i(\sigma^{-1}R,\sigma^{-1}R) = 0 \] for \( i > 0 \)) then every bounded f.g. projective \( \sigma^{-1}R \)-module chain complex \( D \) with \[ [D] \in \operatorname{im}\bigl(K_0(R) \to K_0(\sigma^{-1}R)\bigr) \] is chain equivalent to \( \sigma^{-1}C \) for a bounded f.g. projective \( R \)-module chain complex \( C \), and that there is a localization exact sequence in higher algebraic \( K \)-theory \[ \dots \to K_n(R) \to K_n(\sigma^{-1}R)\to K_n(R,\sigma) \to K_{n-1}(R) \to \dots, \] extending to the left the sequence obtained for \( n \leq 1 \) by Schofield. For a noncommutative localization \( \sigma^{-1}R \) of a ring with involution \( R \) there are analogous results for algebraic \( L \)-theory, extending the results of Vogel from quadratic to symmetric \( L \)-theory.

A. Neeman, A. Ranicki, and A. Schofield: “Representations of algebras as universal localizations,” Math. Proc. Camb. Philos. Soc. 136 : 1 (2004), pp. 105–117. MR 2034017 Zbl 1088.16020 ArXiv math/0205034 article

Given a presentation of a finitely presented group, there is a natural way to represent the group as the fundamental group of a 2-complex. The first part of this paper demonstrates one possible way to represent a finitely presented algebra \( S \) in a similarly compact form. From a presentation of the algebra, we construct a quiver with relations whose path algebra is finite dimensional. When we adjoin inverses to some of the arrows in the quiver, we show that the path algebra of the new quiver with relations is \( M_n(S) \) where \( n \) is the number of vertices in our quiver. The slogan would be that every finitely presented algebra is Morita equivalent to a universal localization of a finite dimensional algebra.

Aidan H. Schofield	Related
Amnon Neeman	Related

A. Neeman and A. Ranicki: “Noncommutative localisation in algebraic \( K \)-theory, I,” Geom. Topol. 8 (2004), pp. 1385–1425. MR 2119300 Zbl 1083.18007 ArXiv math/0410620 article

This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic \( K \)-theory. The main result goes as follows. Let \( A \) be an associative ring and let \( A \to B \) be the localisation with respect to a set \( \sigma \) of maps between finitely generated projective \( A \)-modules. Suppose that \( \mathrm{Tor}_n^A(B,B) \) vanishes for all \( n > 0 \). View each map in \( \sigma \) as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes \( D\operatorname{perf}(A) \). Denote by \( \langle\sigma\rangle \) the thick subcategory generated by these complexes. Then the canonical functor \[ D\operatorname{perf}(A) \to D\operatorname{perf}(B) \] induces (up to direct factors) an equivalence \[ D\operatorname{perf}(A)/\langle\sigma\rangle \to D\operatorname{perf}(B) .\] As a consequence, one obtains a homotopy fibre sequence \[ K(A,\sigma) \to K(A) \to K(B) \] (up to surjectivity of \( K_0(A) \to K_0(B) \)) of Waldhausen \( K \)-theory spectra.

In subsequent articles we will present the \( K \)- and \( L \)-theoretic consequences of the main theorem in a form more suitable for the applications to surgery. For example if, in addition to the vanishing of \( \mathrm{Tor}_n^A(B,B) \), we also assume that every map in \( \sigma \) is a monomorphism, then there is a description of the homotopy fiber of the map \( K(A) \to K(B) \) as the Quillen \( K \)-theory of a suitable exact category of torsion modules.

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