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.
Author(s): Bruce A. Magurn
Series: Encyclopedia of Mathematics and its Applications
Edition: 1st
Publisher: Cambridge University Press
Year: 2002
Language: English
Pages: 690
Contents......Page all_4077_to_00690.cpc0007.djvu
Preface......Page all_4077_to_00690.cpc0011.djvu
0 Preliminaries......Page all_4077_to_00690.cpc0015.djvu
I Groups of Modules: K0......Page all_4077_to_00690.cpc0029.djvu
1A Bases......Page all_4077_to_00690.cpc0031.djvu
1B Matrix Representations......Page all_4077_to_00690.cpc0040.djvu
1C Absence of Dimension......Page all_4077_to_00690.cpc0052.djvu
2A Direct Summands......Page all_4077_to_00690.cpc0057.djvu
2B Summands of Free Modules......Page all_4077_to_00690.cpc0065.djvu
3A Semigroups of Isomorphism Classes......Page all_4077_to_00690.cpc0071.djvu
3B Semigroups to Groups......Page all_4077_to_00690.cpc0085.djvu
3C Grothendieck Groups......Page all_4077_to_00690.cpc0097.djvu
3D Resolutions......Page all_4077_to_00690.cpc0109.djvu
4A Adding Copies of R......Page all_4077_to_00690.cpc0118.djvu
4B Stably Free Modules......Page all_4077_to_00690.cpc0122.djvu
4C When Stably Free Modules Are Free......Page all_4077_to_00690.cpc0127.djvu
4D Stable Rank......Page all_4077_to_00690.cpc0134.djvu
4E Dimensions of a Ring......Page all_4077_to_00690.cpc0142.djvu
5A Semirings......Page all_4077_to_00690.cpc0147.djvu
5B Burnside Rings......Page all_4077_to_00690.cpc0149.djvu
5C Tensor Products of Modules......Page all_4077_to_00690.cpc0155.djvu
6A K0 of Related Rings......Page all_4077_to_00690.cpc0174.djvu
6B Go of Related Rings......Page all_4077_to_00690.cpc0183.djvu
6C K_n as a Functor......Page all_4077_to_00690.cpc0188.djvu
6D The Jacobson Radical......Page all_4077_to_00690.cpc0192.djvu
6E Localization......Page all_4077_to_00690.cpc0199.djvu
II Sources of K_0......Page all_4077_to_00690.cpc0217.djvu
7A Algebraic Integers......Page all_4077_to_00690.cpc0219.djvu
7B Dedekind Domains......Page all_4077_to_00690.cpc0226.djvu
7C Ideal Class Groups......Page all_4077_to_00690.cpc0238.djvu
7D Extensions and Norms......Page all_4077_to_00690.cpc0244.djvu
7E K_0 and G_0 of Dedekind Domains......Page all_4077_to_00690.cpc0256.djvu
8A Linear Representations......Page all_4077_to_00690.cpc0266.djvu
8B Representing Finite Groups Over Fields......Page all_4077_to_00690.cpc0279.djvu
8C Semisimple Rings......Page all_4077_to_00690.cpc0291.djvu
8D Characters......Page all_4077_to_00690.cpc0314.djvu
III Groups of Matrices: K_1......Page all_4077_to_00690.cpc0331.djvu
9A Elementary Matrices......Page all_4077_to_00690.cpc0333.djvu
9B Commutators and K_1(R)......Page all_4077_to_00690.cpc0336.djvu
9C Determinants......Page all_4077_to_00690.cpc0342.djvu
9D The Bass K_1 of a Category......Page all_4077_to_00690.cpc0347.djvu
10 Stability for K_1(R)......Page all_4077_to_00690.cpc0356.djvu
10A Surjective Stability......Page all_4077_to_00690.cpc0357.djvu
10B Infective Stability......Page all_4077_to_00690.cpc0362.djvu
11A Congruence Subgroups of GL_n(R)......Page all_4077_to_00690.cpc0371.djvu
11B Congruence Subgroups of SL_n(R)......Page all_4077_to_00690.cpc0383.djvu
11C Mennicke Symbols......Page all_4077_to_00690.cpc0388.djvu
IV Relations Among Matrices: K_2......Page all_4077_to_00690.cpc0413.djvu
12A Definition and Properties of K_2(R)......Page all_4077_to_00690.cpc0415.djvu
12B Elements of St(R) and K_2(R)......Page all_4077_to_00690.cpc0427.djvu
13 Exact Sequences......Page all_4077_to_00690.cpc0444.djvu
13A The Relative Sequence......Page all_4077_to_00690.cpc0445.djvu
13B Excision and the Mayer-Vietoris Sequence......Page all_4077_to_00690.cpc0470.djvu
13C The Localization Sequence......Page all_4077_to_00690.cpc0495.djvu
14 Universal Algebras......Page all_4077_to_00690.cpc0502.djvu
14A Presentation of Algebras......Page all_4077_to_00690.cpc0503.djvu
14B Graded Rings......Page all_4077_to_00690.cpc0507.djvu
14C The Tensor Algebra......Page all_4077_to_00690.cpc0511.djvu
14D Symmetric and Exterior Algebras......Page all_4077_to_00690.cpc0519.djvu
14E The Milnor Ring......Page all_4077_to_00690.cpc0532.djvu
14F Tame Symbols......Page all_4077_to_00690.cpc0548.djvu
14G Norms on Milnor K-Theory......Page all_4077_to_00690.cpc0561.djvu
14H Matsumoto's Theorem......Page all_4077_to_00690.cpc0571.djvu
V Sources of K_2......Page all_4077_to_00690.cpc0581.djvu
15A Hilbert Symbols......Page all_4077_to_00690.cpc0583.djvu
15B Metric Completion of Fields......Page all_4077_to_00690.cpc0586.djvu
15C The p-Adic Numbers and Quadratic Reciprocity......Page all_4077_to_00690.cpc0594.djvu
15D Local Fields and Norm Residue Symbols......Page all_4077_to_00690.cpc0609.djvu
16A The Brauer Group of a Field......Page all_4077_to_00690.cpc0624.djvu
16B Splitting Fields......Page all_4077_to_00690.cpc0637.djvu
16C Twisted Group Rings......Page all_4077_to_00690.cpc0643.djvu
16D The K_2 Connection......Page all_4077_to_00690.cpc0650.djvu
A Sets, Classes, Functions......Page all_4077_to_00690.cpc0659.djvu
B Chain Conditions, Composition Series......Page all_4077_to_00690.cpc0661.djvu
Special Symbols......Page all_4077_to_00690.cpc0671.djvu
References......Page all_4077_to_00690.cpc0675.djvu
Index......Page all_4077_to_00690.cpc0685.djvu
