Algebraic K-TheorySpringer Science & Business Media, 2013 M03 14 - 440 pages Algebraic K-theory is a modern branch of algebra which has many important applications in fundamental areas of mathematics connected with algebra, topology, algebraic geometry, functional analysis and algebraic number theory. Methods of algebraic K-theory are actively used in algebra and related fields, achieving interesting results. This book presents the elements of algebraic K-theory, based essentially on the fundamental works of Milnor, Swan, Bass, Quillen, Karoubi, Gersten, Loday and Waldhausen. It includes all principal algebraic K-theories, connections with topological K-theory and cyclic homology, applications to the theory of monoid and polynomial algebras and in the theory of normed algebras. This volume will be of interest to graduate students and research mathematicians who want to learn more about K-theory. |
Contents
The BassWhitehead functor | 29 |
The Milnor functor | 35 |
Higher Kfunctors | 43 |
35 | 65 |
43 | 72 |
C | 96 |
F Transfer map in the localization theorem | 117 |
KTheory of Swan | 127 |
Products in algebraic Ktheory | 222 |
Stability | 248 |
Connection of Quillens plus construction with Swans algebraic | 270 |
Comparison of Swans and KaroubiVillamayors algebraic | 278 |
Relation between algebraic and topological Ktheories | 289 |
Ktheory of special normed algebras and Z2graded Calgebras | 305 |
Isomorphism of Swans and KaroubiVillamayors Ktheories | 353 |
The problem of Serre for polynomial and monoid alge | 361 |
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Common terms and phrases
A-fibration A-homomorphism A₁ abelian group admissible sequence algebraic K-theory arbitrary b₁ Banach k-algebra basepoint BGL(A bijection C*-algebra C*-algebra with unit canonical map cartesian square category with cofibrations Chapter clear cofibrations commutative diagram commutative ring Consider continuous map Corollary cotriple cyclic homology defined DEFINITION denote domain epimorphism exact category exact functor exists fibration fibre finitely generated projective full subcategory given GL(A hence homomorphism homotopy equivalence implies injective Int(M K-functors K-groups Karoubi-Villamayor's Ko(A Lemma Let f Let G M₁ matrix maximal ideal Modf modules monoid morphism Noetherian normal object obtain P₁ pairs polyhedron polynomial projective A-module projective modules PROOF OF THEOREM Proposition prove Quillen's r₁ raise the norm respectively ring with unit satisfies seminormal short exact sequence simplicial ring Spec(R special normed R-algebra St(A subgroup submonoid subset surjective Swan's torsion free trivial Z2-graded C*-algebras