Noncommutative localisation in algebraic K-theory I
By A. Neeman, A. Ranicki. 2004
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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 → B be the localisation with respect to a set σ of maps between finitely generated projective A-modules. Suppose that Tor A n (B,B) vanishes for all n > 0. View each map in a as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes D p e r f (A). Denote by the thick subcategory generated by these complexes. Then the canonical functor D p e r f (A) → D p e r f (B) induces (up to direct factors) an equivalence D p e r f (A)/ → D p e r f (B). As a consequence, one obtains a homotopy fibre sequence K(A,σ) → K(A) → K(B) (up to surjectivity of K 0 (A) → K 0 (B)) of Waldhausen K-theory spectra. In subsequent articles [26, 27] 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 Tor A n (B,B), we also assume that every map in σ is a monomorphism, then there is a description of the homotopy fiber of the map K(A) → K(B) as the Quillen K-theory of a suitable exact category of torsion modules.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/257e8b4347c429a35ce09e6b41631b39d622dc15
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Citations in the corpus
- Reducing Homological Conjectures by n-Recollements (2014, methodology): “Here, we focus on the recollements of derived categories of algebras which are the generalization of derived equivalences and provide a useful reduction technique for some homological properties such as the finiteness of global dimension [32, 21, 1], the finiteness of finitistic dimension [17, 9] an”