Algebraic K-theory and Algebraic Number Theory

Algebraic K-theory and Algebraic Number Theory
Title Algebraic K-theory and Algebraic Number Theory PDF eBook
Author Michael R. Stein
Publisher American Mathematical Soc.
Pages 506
Release 1989
Genre Mathematics
ISBN 0821850903

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This volume contains the proceedings of a seminar on Algebraic $K$-theory and Algebraic Number Theory, held at the East-West Center in Honolulu in January 1987. The seminar, which hosted nearly 40 experts from the U.S. and Japan, was motivated by the wide range of connections between the two topics, as exemplified in the work of Merkurjev, Suslin, Beilinson, Bloch, Ramakrishnan, Kato, Saito, Lichtenbaum, Thomason, and Ihara. As is evident from the diversity of topics represented in these proceedings, the seminar provided an opportunity for mathematicians from both areas to initiate further interactions between these two areas.

Algebraic K-Theory and Its Applications

Algebraic K-Theory and Its Applications
Title Algebraic K-Theory and Its Applications PDF eBook
Author Jonathan Rosenberg
Publisher Springer Science & Business Media
Pages 404
Release 2012-12-06
Genre Mathematics
ISBN 1461243149

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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.

The $K$-book

The $K$-book
Title The $K$-book PDF eBook
Author Charles A. Weibel
Publisher American Mathematical Soc.
Pages 634
Release 2013-06-13
Genre Mathematics
ISBN 0821891324

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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

Algebraic K-Theory
Title Algebraic K-Theory PDF eBook
Author Vasudevan Srinivas
Publisher Springer Science & Business Media
Pages 328
Release 2013-11-21
Genre Science
ISBN 1489967354

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Introduction to Algebraic K-theory

Introduction to Algebraic K-theory
Title Introduction to Algebraic K-theory PDF eBook
Author John Willard Milnor
Publisher Princeton University Press
Pages 204
Release 1971
Genre Mathematics
ISBN 9780691081014

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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.

Transformation Groups and Algebraic K-Theory

Transformation Groups and Algebraic K-Theory
Title Transformation Groups and Algebraic K-Theory PDF eBook
Author Wolfgang Lück
Publisher Springer
Pages 455
Release 2006-11-14
Genre Mathematics
ISBN 3540468277

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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.

The Local Structure of Algebraic K-Theory

The Local Structure of Algebraic K-Theory
Title The Local Structure of Algebraic K-Theory PDF eBook
Author Bjørn Ian Dundas
Publisher Springer Science & Business Media
Pages 447
Release 2012-09-06
Genre Mathematics
ISBN 1447143930

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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.