Introduction To Algebraic K Theory

Author : John Milnor
Publisher : Princeton University Press
Page : 200 pages
File Size : 51,5 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400881796

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Introduction to Algebraic K-Theory. (AM-72), Volume 72 by John Milnor Pdf

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 : 49,9 Mb
Release : 2013-06-13
Category : Mathematics
ISBN : 9780821891322

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The $K$-book by Charles A. Weibel Pdf

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 : Bruce A. Magurn
Publisher : Cambridge University Press
Page : 702 pages
File Size : 53,7 Mb
Release : 2002-05-20
Category : Mathematics
ISBN : 0521800781

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An Algebraic Introduction to K-Theory by Bruce A. Magurn Pdf

An introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra.

Author : Jonathan Rosenberg
Publisher : Springer Science & Business Media
Page : 404 pages
File Size : 51,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461243144

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Algebraic K-Theory and Its Applications by Jonathan Rosenberg Pdf

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 : Vasudevan Srinivas
Publisher : Springer Science & Business Media
Page : 328 pages
File Size : 55,6 Mb
Release : 2013-11-21
Category : Science
ISBN : 9781489967350

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Algebraic K-Theory by Vasudevan Srinivas Pdf

Author : Max Karoubi
Publisher : Springer Science & Business Media
Page : 337 pages
File Size : 54,6 Mb
Release : 2009-11-27
Category : Mathematics
ISBN : 9783540798903

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K-Theory by Max Karoubi Pdf

From the Preface: K-theory was introduced by A. Grothendieck in his formulation of the Riemann- Roch theorem. For each projective algebraic variety, Grothendieck constructed a group from the category of coherent algebraic sheaves, and showed that it had many nice properties. Atiyah and Hirzebruch considered a topological analog defined for any compact space X, a group K{X) constructed from the category of vector bundles on X. It is this ''topological K-theory" that this book will study. Topological K-theory has become an important tool in topology. Using K- theory, Adams and Atiyah were able to give a simple proof that the only spheres which can be provided with H-space structures are S1, S3 and S7. Moreover, it is possible to derive a substantial part of stable homotopy theory from K-theory. The purpose of this book is to provide advanced students and mathematicians in other fields with the fundamental material in this subject. In addition, several applications of the type described above are included. In general we have tried to make this book self-contained, beginning with elementary concepts wherever possible; however, we assume that the reader is familiar with the basic definitions of homotopy theory: homotopy classes of maps and homotopy groups.Thus this book might be regarded as a fairly self-contained introduction to a "generalized cohomology theory".

Author : John Willard Milnor
Publisher : Princeton University Press
Page : 204 pages
File Size : 45,6 Mb
Release : 1971
Category : Mathematics
ISBN : 0691081018

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Introduction to Algebraic K-theory by John Willard Milnor Pdf

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 : Bjørn Ian Dundas,Thomas G. Goodwillie,Randy McCarthy
Publisher : Springer Science & Business Media
Page : 447 pages
File Size : 42,7 Mb
Release : 2012-09-06
Category : Mathematics
ISBN : 9781447143932

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The Local Structure of Algebraic K-Theory by Bjørn Ian Dundas,Thomas G. Goodwillie,Randy McCarthy Pdf

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 : Richard G. Swan
Publisher : Springer
Page : 269 pages
File Size : 54,8 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540359173

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Algebraic K-Theory by Richard G. Swan Pdf

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 : M. Rørdam,Flemming Larsen,N. Laustsen
Publisher : Cambridge University Press
Page : 260 pages
File Size : 54,8 Mb
Release : 2000-07-20
Category : Mathematics
ISBN : 0521789443

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An Introduction to K-Theory for C*-Algebras by M. Rørdam,Flemming Larsen,N. Laustsen Pdf

This book provides a very elementary introduction to K-theory for C*-algebras, and is ideal for beginning graduate students.

Author : John R. Silvester
Publisher : Chapman & Hall
Page : 274 pages
File Size : 49,7 Mb
Release : 1981
Category : Mathematics
ISBN : UOM:39015017339980

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Introduction to Algebraic K-theory by John R. Silvester Pdf

Author : Uwe Jannsen
Publisher : Springer
Page : 260 pages
File Size : 42,8 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540469414

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Mixed Motives and Algebraic K-Theory by Uwe Jannsen Pdf

The relations that could or should exist between algebraic cycles, algebraic K-theory, and the cohomology of - possibly singular - varieties, are the topic of investigation of this book. The author proceeds in an axiomatic way, combining the concepts of twisted Poincaré duality theories, weights, and tensor categories. One thus arrives at generalizations to arbitrary varieties of the Hodge and Tate conjectures to explicit conjectures on l-adic Chern characters for global fields and to certain counterexamples for more general fields. It is to be hoped that these relations ions will in due course be explained by a suitable tensor category of mixed motives. An approximation to this is constructed in the setting of absolute Hodge cycles, by extending this theory to arbitrary varieties. The book can serve both as a guide for the researcher, and as an introduction to these ideas for the non-expert, provided (s)he knows or is willing to learn about K-theory and the standard cohomology theories of algebraic varieties.

Author : John Guaschi,Daniel Juan-Pineda,Silvia Millán López
Publisher : Springer
Page : 80 pages
File Size : 41,5 Mb
Release : 2018-11-03
Category : Mathematics
ISBN : 9783319994895

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The Lower Algebraic K-Theory of Virtually Cyclic Subgroups of the Braid Groups of the Sphere and of ZB4(S2) by John Guaschi,Daniel Juan-Pineda,Silvia Millán López Pdf

This volume deals with the K-theoretical aspects of the group rings of braid groups of the 2-sphere. The lower algebraic K-theory of the finite subgroups of these groups up to eleven strings is computed using a wide variety of tools. Many of the techniques extend to the general case, and the results reveal new K-theoretical phenomena with respect to the previous study of other families of groups. The second part of the manuscript focusses on the case of the 4-string braid group of the 2-sphere, which is shown to be hyperbolic in the sense of Gromov. This permits the computation of the infinite maximal virtually cyclic subgroups of this group and their conjugacy classes, and applying the fact that this group satisfies the Fibred Isomorphism Conjecture of Farrell and Jones, leads to an explicit calculation of its lower K-theory. Researchers and graduate students working in K-theory and surface braid groups will constitute the primary audience of the manuscript, particularly those interested in the Fibred Isomorphism Conjecture, and the computation of Nil groups and the lower algebraic K-groups of group rings. The manuscript will also provide a useful resource to researchers who wish to learn the techniques needed to calculate lower algebraic K-groups, and the bibliography brings together a large number of references in this respect.

Author : Michael Atiyah
Publisher : CRC Press
Page : 138 pages
File Size : 52,6 Mb
Release : 2018-03-05
Category : Mathematics
ISBN : 9780429973178

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K-theory by Michael Atiyah Pdf

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.

Author : Serge Lang
Publisher : Courier Dover Publications
Page : 273 pages
File Size : 52,7 Mb
Release : 2019-03-20
Category : Mathematics
ISBN : 9780486839806

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Introduction to Algebraic Geometry by Serge Lang Pdf

Author Serge Lang defines algebraic geometry as the study of systems of algebraic equations in several variables and of the structure that one can give to the solutions of such equations. The study can be carried out in four ways: analytical, topological, algebraico-geometric, and arithmetic. This volume offers a rapid, concise, and self-contained introductory approach to the algebraic aspects of the third method, the algebraico-geometric. The treatment assumes only familiarity with elementary algebra up to the level of Galois theory. Starting with an opening chapter on the general theory of places, the author advances to examinations of algebraic varieties, the absolute theory of varieties, and products, projections, and correspondences. Subsequent chapters explore normal varieties, divisors and linear systems, differential forms, the theory of simple points, and algebraic groups, concluding with a focus on the Riemann-Roch theorem. All the theorems of a general nature related to the foundations of the theory of algebraic groups are featured.