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Author : Hyman Bass
Publisher : American Mathematical Soc.
Page : 68 pages
File Size : 44,5 Mb
Release : 1974-12-31
Category : Mathematics
ISBN : 9780821816707
Author : Bjørn Ian Dundas,Thomas G. Goodwillie,Randy McCarthy
Publisher : Springer Science & Business Media
Page : 447 pages
File Size : 54,6 Mb
Release : 2012-09-06
Category : Mathematics
ISBN : 9781447143932
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
Page : 204 pages
File Size : 44,9 Mb
Release : 1971
Category : Mathematics
ISBN : 0691081018
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 : John Milnor
Publisher : Princeton University Press
Page : 200 pages
File Size : 46,5 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400881796
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 : Charles A. Weibel
Publisher : American Mathematical Soc.
Page : 634 pages
File Size : 42,9 Mb
Release : 2013-06-13
Category : Mathematics
ISBN : 9780821891322
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 : Richard G. Swan
Publisher : Springer
Page : 269 pages
File Size : 40,8 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540359173
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 : Hvedri Inassaridze
Publisher : Springer Science & Business Media
Page : 444 pages
File Size : 46,5 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9789401585699
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 : John R. Silvester
Publisher : Chapman & Hall
Page : 274 pages
File Size : 41,5 Mb
Release : 1981
Category : Mathematics
ISBN : UOM:39015017339980
Author : Eric Friedlander,Daniel R. Grayson
Publisher : Springer Science & Business Media
Page : 1148 pages
File Size : 40,8 Mb
Release : 2005-07-18
Category : Mathematics
ISBN : 9783540230199
This handbook offers a compilation of techniques and results in K-theory. Each chapter is dedicated to a specific topic and is written by a leading expert. Many chapters present historical background; some present previously unpublished results, whereas some present the first expository account of a topic; many discuss future directions as well as open problems. It offers an exposition of our current state of knowledge as well as an implicit blueprint for future research.
Author : Bruce A. Magurn
Publisher : Cambridge University Press
Page : 704 pages
File Size : 41,6 Mb
Release : 2002-05-20
Category : Mathematics
ISBN : 9781107079441
This is an introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain). The presentation is almost entirely self-contained, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. No experience with analysis, geometry, number theory or topology is assumed. Within the context of linear algebra, K-theory organises and clarifies the relations among ideal class groups, group representations, quadratic forms, dimensions of a ring, determinants, quadratic reciprocity and Brauer groups of fields. By including introductions to standard algebra topics (tensor products, localisation, Jacobson radical, chain conditions, Dedekind domains, semi-simple rings, exterior algebras), the author makes algebraic K-theory accessible to first-year graduate students and other mathematically sophisticated readers. Even if your algebra is rusty, you can read this book; the necessary background is here, with proofs.
Author : Bruce A. Magurn
Publisher : Cambridge University Press
Page : 702 pages
File Size : 53,6 Mb
Release : 2002-05-20
Category : Mathematics
ISBN : 0521800781
An introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra.
Author : Eric Friedlander,M. R. Stein
Publisher : Springer
Page : 526 pages
File Size : 41,9 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540386469
Author : Emilio Lluis-Puebla,Jean-Louis Loday,Henri Gillet,Christophe Soule,Victor Snaith
Publisher : Springer
Page : 172 pages
File Size : 43,5 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540466390
This book is a general introduction to Higher Algebraic K-groups of rings and algebraic varieties, which were first defined by Quillen at the beginning of the 70's. These K-groups happen to be useful in many different fields, including topology, algebraic geometry, algebra and number theory. The goal of this volume is to provide graduate students, teachers and researchers with basic definitions, concepts and results, and to give a sampling of current directions of research. Written by five specialists of different parts of the subject, each set of lectures reflects the particular perspective ofits author. As such, this volume can serve as a primer (if not as a technical basic textbook) for mathematicians from many different fields of interest.
Author : Daniel Scott Farley,Ivonne Johanna Ortiz
Publisher : Springer
Page : 148 pages
File Size : 50,6 Mb
Release : 2014-08-27
Category : Mathematics
ISBN : 9783319081533
The Farrell-Jones isomorphism conjecture in algebraic K-theory offers a description of the algebraic K-theory of a group using a generalized homology theory. In cases where the conjecture is known to be a theorem, it gives a powerful method for computing the lower algebraic K-theory of a group. This book contains a computation of the lower algebraic K-theory of the split three-dimensional crystallographic groups, a geometrically important class of three-dimensional crystallographic group, representing a third of the total number. The book leads the reader through all aspects of the calculation. The first chapters describe the split crystallographic groups and their classifying spaces. Later chapters assemble the techniques that are needed to apply the isomorphism theorem. The result is a useful starting point for researchers who are interested in the computational side of the Farrell-Jones isomorphism conjecture, and a contribution to the growing literature in the field.
Author : Hvedri Inassaridze
Publisher : Springer
Page : 324 pages
File Size : 51,6 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540471622