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An Algebraic Introduction to K-Theory : Encyclopedia of Mathematics and Its Applications - Bruce A. Magurn

An Algebraic Introduction to K-Theory

Encyclopedia of Mathematics and Its Applications


Published: 20th May 2002
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This book is both an introduction to K-theory and a text in algebra. These two roles are entirely compatible. On the one hand, nothing more than the basic algebra of groups, rings, and modules is needed to explain the clasical algebraic K-theory. On the other hand, K-theory is a natural organizing principle for the standard topics of a second course in algebra, and these topics are presented carefully here. The reader will not only learn algebraic K-theory, but also Dedekind domains, class groups, semisimple rings, character theory, quadratic forms, tensor products, localization, completion, tensor algebras, symmetric algebras, exterior algebras, central simple algebras, and Brauer groups. The presentation is self-contained, with all the necessary background and proofs, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. The prerequisites are minimal: just a first semester of algebra (including Galois theory and modules over a principal ideal domain). No experience with homological algebra, analysis, geometry, number theory, or topology is assumed. The author has successfuly used this text to teach algebra to first year graduate students. Selected topics can be used to construct a variety of one-semester courses; coverage of the entire text requires a full year.

'... this is a well written introduction to the theory of the algebraic K-groups Ko, K1 and K2; the author has done a wonderful job in presenting the material in a clear way that will be accessible to readers with a modest background in algebra.' Franz Lemmermeyer, Zentralblatt MATH 'This is a fine introduction to algebraic K-theory, requiring only a basic preliminary knowledge of groups, rings and modules.' European Mathematical Society '... a fine introduction to algebraic K-theory ...'. EMS Newsletter '... an excellent introduction to the algebraic K-theory.' Proceedings of the Edinburgh Mathematical Society

Prefacep. xi
Preliminariesp. 1
Groups of Modules: K[subscript 0]p. 15
Free Modulesp. 17
Basesp. 17
Matrix Representationsp. 26
Absence of Dimensionp. 38
Projective Modulesp. 43
Direct Summandsp. 43
Summands of Free Modulesp. 51
Grothendieck Groupsp. 57
Semigroups of Isomorphism Classesp. 57
Semigroups to Groupsp. 71
Grothendieck Groupsp. 83
Resolutionsp. 95
Stability for Projective Modulesp. 104
Adding Copies of Rp. 104
Stably Free Modulesp. 108
When Stably Free Modules Are Freep. 113
Stable Rankp. 120
Dimensions of a Ringp. 128
Multiplying Modulesp. 133
Semiringsp. 133
Burnside Ringsp. 135
Tensor Products of Modulesp. 141
Change of Ringsp. 160
K[subscript 0] of Related Ringsp. 160
G[subscript 0] of Related Ringsp. 169
K[subscript 0] as a Functorp. 174
The Jacobson Radicalp. 178
Localizationp. 185
Sources of K[subscript 0]p. 203
Number Theoryp. 205
Algebraic Integersp. 205
Dedekind Domainsp. 212
Ideal Class Groupsp. 224
Extensions and Normsp. 230
K[subscript 0] and G[subscript 0] of Dedekind Domainsp. 242
Group Representation Theoryp. 252
Linear Representationsp. 252
Representing Finite Groups Over Fieldsp. 265
Semisimple Ringsp. 277
Charactersp. 300
Groups of Matrices: K[subscript 1]p. 317
Definition of K[subscript 1]p. 319
Elementary Matricesp. 319
Commutators and K[subscript 1](R)p. 322
Determinantsp. 328
The Bass K[subscript 1] of a Categoryp. 333
Stability for K[subscript 1](R)p. 342
Surjective Stabilityp. 343
Injective Stabilityp. 348
Relative K[subscript 1]p. 357
Congruence Subgroups of GL[subscript n](R)p. 357
Congruence Subgroups of SL[subscript n](R)p. 369
Mennicke Symbolsp. 374
Relations Among Matrices: K[subscript 2]p. 399
K[subscript 2](R) and Steinberg Symbolsp. 401
Definition and Properties of K[subscript 2](R)p. 401
Elements of St(R) and K[subscript 2](R)p. 413
Exact Sequencesp. 430
The Relative Sequencep. 431
Excision and the Mayer-Vietoris Sequencep. 456
The Localization Sequencep. 481
Universal Algebrasp. 488
Presentation of Algebrasp. 489
Graded Ringsp. 493
The Tensor Algebrap. 497
Symmetric and Exterior Algebrasp. 505
The Milnor Ringp. 518
Tame Symbolsp. 534
Norms on Milnor K-Theoryp. 547
Matsumoto's Theoremp. 557
Sources of K[subscript 2]p. 567
Symbols in Arithmeticp. 569
Hilbert Symbolsp. 569
Metric Completion of Fieldsp. 572
The p-Adic Numbers and Quadratic Reciprocityp. 580
Local Fields and Norm Residue Symbolsp. 595
Brauer Groupsp. 610
The Brauer Group of a Fieldp. 610
Splitting Fieldsp. 623
Twisted Group Ringsp. 629
The K[subscript 2] Connectionp. 636
Appendixp. 645
Sets, Classes, Functionsp. 645
Chain Conditions, Composition Seriesp. 647
Special Symbolsp. 657
Referencesp. 661
Indexp. 671
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521800785
ISBN-10: 0521800781
Series: Encyclopedia of Mathematics and Its Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 692
Published: 20th May 2002
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 23.9 x 16.5  x 3.9
Weight (kg): 1.14