Introduction to Some Methods of Algebraic $K$-Theory
Author: Hyman Bass
Publisher: American Mathematical Soc.
Published: 1974-12-31
Total Pages: 78
ISBN-13: 0821816705
DOWNLOAD EBOOK →Author: Hyman Bass
Publisher: American Mathematical Soc.
Published: 1974-12-31
Total Pages: 78
ISBN-13: 0821816705
DOWNLOAD EBOOK →Author: Charles A. Weibel
Publisher: American Mathematical Soc.
Published: 2013-06-13
Total Pages: 634
ISBN-13: 0821891324
DOWNLOAD EBOOK →Informally, $K$-theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebr
Author: Bjørn Ian Dundas
Publisher: Springer Science & Business Media
Published: 2012-09-06
Total Pages: 447
ISBN-13: 1447143930
DOWNLOAD EBOOK →Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry. As is commonly encountered, this powerful mathematical object is very hard to calculate. Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few complete calculations were available before the discovery of homological invariants offered by motivic cohomology and topological cyclic homology. This book covers the connection between algebraic K-theory and Bökstedt, Hsiang and Madsen's topological cyclic homology and proves that the difference between the theories are ‘locally constant’. The usefulness of this theorem stems from being more accessible for calculations than K-theory, and hence a single calculation of K-theory can be used with homological calculations to obtain a host of ‘nearby’ calculations in K-theory. For instance, Quillen's calculation of the K-theory of finite fields gives rise to Hesselholt and Madsen's calculations for local fields, and Voevodsky's calculations for the integers give insight into the diffeomorphisms of manifolds. In addition to the proof of the full integral version of the local correspondence between K-theory and topological cyclic homology, the book provides an introduction to the necessary background in algebraic K-theory and highly structured homotopy theory; collecting all necessary tools into one common framework. It relies on simplicial techniques, and contains an appendix summarizing the methods widely used in the field. The book is intended for graduate students and scientists interested in algebraic K-theory, and presupposes a basic knowledge of algebraic topology.
Author: John Willard Milnor
Publisher: Princeton University Press
Published: 1971
Total Pages: 204
ISBN-13: 9780691081014
DOWNLOAD EBOOK →Algebraic K-theory describes a branch of algebra that centers about two functors. K0 and K1, which assign to each associative ring ∧ an abelian group K0∧ or K1∧ respectively. Professor Milnor sets out, in the present work, to define and study an analogous functor K2, also from associative rings to abelian groups. Just as functors K0 and K1 are important to geometric topologists, K2 is now considered to have similar topological applications. The exposition includes, besides K-theory, a considerable amount of related arithmetic.
Author: Bruce A. Magurn
Publisher: Cambridge University Press
Published: 2002-05-20
Total Pages: 702
ISBN-13: 9780521800785
DOWNLOAD EBOOK →An introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra.
Author: Hvedri Inassaridze
Publisher: Springer Science & Business Media
Published: 2013-03-14
Total Pages: 444
ISBN-13: 9401585695
DOWNLOAD EBOOK →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.
Author: Jonathan Rosenberg
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 404
ISBN-13: 1461243149
DOWNLOAD EBOOK →Algebraic K-Theory is crucial in many areas of modern mathematics, especially algebraic topology, number theory, algebraic geometry, and operator theory. This text is designed to help graduate students in other areas learn the basics of K-Theory and get a feel for its many applications. Topics include algebraic topology, homological algebra, algebraic number theory, and an introduction to cyclic homology and its interrelationship with K-Theory.
Author: Richard G. Swan
Publisher: Springer
Published: 2006-11-14
Total Pages: 269
ISBN-13: 3540359176
DOWNLOAD EBOOK →From the Introduction: "These notes are taken from a course on algebraic K-theory [given] at the University of Chicago in 1967. They also include some material from an earlier course on abelian categories, elaborating certain parts of Gabriel's thesis. The results on K-theory are mostly of a very general nature."
Author: John R. Silvester
Publisher: Chapman & Hall
Published: 1981
Total Pages: 274
ISBN-13:
DOWNLOAD EBOOK →Author: John Milnor
Publisher: Princeton University Press
Published: 2016-03-02
Total Pages: 200
ISBN-13: 140088179X
DOWNLOAD EBOOK →Algebraic K-theory describes a branch of algebra that centers about two functors. K0 and K1, which assign to each associative ring ∧ an abelian group K0∧ or K1∧ respectively. Professor Milnor sets out, in the present work, to define and study an analogous functor K2, also from associative rings to abelian groups. Just as functors K0 and K1 are important to geometric topologists, K2 is now considered to have similar topological applications. The exposition includes, besides K-theory, a considerable amount of related arithmetic.