Algebraic K Theory

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The K Book

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Author : Charles A. Weibel
language : en
Publisher: American Mathematical Soc.
Release Date : 2013-06-13

The K Book written by Charles A. Weibel and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-13 with Mathematics categories.

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

Algebraic K Theory And Its Applications

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Author : Jonathan Rosenberg
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Algebraic K Theory And Its Applications written by Jonathan Rosenberg and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

Algebraic K-Theory plays an important role in many areas of modern mathematics: most notably algebraic topology, number theory, and algebraic geometry, but even including operator theory. The broad range of these topics has tended to give the subject an aura of inapproachability. This book, based on a course at the University of Maryland in the fall of 1990, is intended to enable graduate students or mathematicians working in other areas not only to learn the basics of algebraic K-Theory, but also to get a feel for its many applications. The required prerequisites are only the standard one-year graduate algebra course and the standard introductory graduate course on algebraic and geometric topology. Many topics from algebraic topology, homological algebra, and algebraic number theory are developed as needed. The final chapter gives a concise introduction to cyclic homology and its interrelationship with K-Theory.

Transformation Groups And Algebraic K Theory

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Author : Wolfgang Lück
language : en
Publisher: Springer
Release Date : 2006-11-14

Transformation Groups And Algebraic K Theory written by Wolfgang Lück and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.

The book focuses on the relation between transformation groups and algebraic K-theory. The general pattern is to assign to a geometric problem an invariant in an algebraic K-group which determines the problem. The algebraic K-theory of modules over a category is studied extensively and appplied to the fundamental category of G-space. Basic details of the theory of transformation groups sometimes hard to find in the literature, are collected here (Chapter I) for the benefit of graduate students. Chapters II and III contain advanced new material of interest to researchers working in transformation groups, algebraic K-theory or related fields.

An Algebraic Introduction To K Theory

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Author : Bruce A. Magurn
language : en
Publisher: Cambridge University Press
Release Date : 2002-05-20

An Algebraic Introduction To K Theory written by Bruce A. Magurn and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002-05-20 with Mathematics categories.

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.

Algebraic K Theory

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Author : Vasudevan Srinivas
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-21

Algebraic K Theory written by Vasudevan Srinivas and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-21 with Social Science categories.

The Local Structure Of Algebraic K Theory

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Author : Bjørn Ian Dundas
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-09-06

The Local Structure Of Algebraic K Theory written by Bjørn Ian Dundas and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-09-06 with Mathematics categories.

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.

Introduction To Algebraic K Theory

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Author : John Milnor
language : en
Publisher: Princeton University Press
Release Date : 2016-03-02

Introduction To Algebraic K Theory written by John Milnor and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-03-02 with Mathematics categories.

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.

Representation Theory And Higher Algebraic K Theory

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Author : Aderemi Kuku
language : en
Publisher: CRC Press
Release Date : 2006-09-27

Representation Theory And Higher Algebraic K Theory written by Aderemi Kuku and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-09-27 with Mathematics categories.

Representation Theory and Higher Algebraic K-Theory is the first book to present higher algebraic K-theory of orders and group rings as well as characterize higher algebraic K-theory as Mackey functors that lead to equivariant higher algebraic K-theory and their relative generalizations. Thus, this book makes computations of higher K-theory of group rings more accessible and provides novel techniques for the computations of higher K-theory of finite and some infinite groups. Authored by a premier authority in the field, the book begins with a careful review of classical K-theory, including clear definitions, examples, and important classical results. Emphasizing the practical value of the usually abstract topological constructions, the author systematically discusses higher algebraic K-theory of exact, symmetric monoidal, and Waldhausen categories with applications to orders and group rings and proves numerous results. He also defines profinite higher K- and G-theory of exact categories, orders, and group rings. Providing new insights into classical results and opening avenues for further applications, the book then uses representation-theoretic techniques-especially induction theory-to examine equivariant higher algebraic K-theory, their relative generalizations, and equivariant homology theories for discrete group actions. The final chapter unifies Farrell and Baum-Connes isomorphism conjectures through Davis-Lück assembly maps.

K Theory

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Author : Michael Atiyah
language : en
Publisher: CRC Press
Release Date : 2018-03-05

K Theory written by Michael Atiyah and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-03-05 with Mathematics categories.

These notes are based on the course of lectures I gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. In fact, rational cohomology is defined in terms of K-theory.The theory is taken as far as the solution of the Hopf invariant problem and a start is mode on the J-homomorphism. In addition to the lecture notes proper, two papers of mine published since 1964 have been reproduced at the end. The first, dealing with operations, is a natural supplement to the material in Chapter III. It provides an alternative approach to operations which is less slick but more fundamental than the Grothendieck method of Chapter III, and it relates operations and filtration. Actually, the lectures deal with compact spaces, not cell-complexes, and so the skeleton-filtration does not figure in the notes. The second paper provides a new approach to K-theory and so fills an obvious gap in the lecture notes.