An algebraic introduction to K-theory / Bruce A. Magurn.

Includes bibliographical references (pages 661-669) and index.

"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.

Selected topics can be used to construct a variety of one-semester courses; coverage of the entire text requires a full year."--Jacket

Part I Groups of Modules: K[subscript 0] 15 -- Chapter 1 Free Modules 17 -- 1A Bases 17 -- 1B Matrix Representations 26 -- 1C Absence of Dimension 38 -- Chapter 2 Projective Modules 43 -- 2A Direct Summands 43 -- 2B Summands of Free Modules 51 -- Chapter 3 Grothendieck Groups 57 -- 3A Semigroups of Isomorphism Classes 57 -- 3B Semigroups to Groups 71 -- 3C Grothendieck Groups 83 -- 3D Resolutions 95 -- Chapter 4 Stability for Projective Modules 104 -- 4A Adding Copies of R 104 -- 4B Stably Free Modules 108 -- 4C When Stably Free Modules Are Free 113 -- 4D Stable Rank 120 -- 4E Dimensions of a Ring 128 -- Chapter 5 Multiplying Modules 133 -- 5A Semirings 133 -- 5B Burnside Rings 135 -- 5C Tensor Products of Modules 141 -- Chapter 6 Change of Rings 160 -- 6A K[subscript 0] of Related Rings 160 -- 6B G[subscript 0] of Related Rings 169 -- 6C K[subscript 0] as a Functor 174 -- 6D The Jacobson Radical 178 -- 6E Localization 185 -- Part II Sources of K[subscript 0] 203 -- Chapter 7 Number Theory 205 -- 7A Algebraic Integers 205 -- 7B Dedekind Domains 212 -- 7C Ideal Class Groups 224 -- 7D Extensions and Norms 230 -- 7E K[subscript 0] and G[subscript 0] of Dedekind Domains 242 -- Chapter 8 Group Representation Theory 252 -- 8A Linear Representations 252 -- 8B Representing Finite Groups Over Fields 265 -- 8C Semisimple Rings 277 -- 8D Characters 300 -- Part III Groups of Matrices: K[subscript 1] 317 -- Chapter 9 Definition of K[subscript 1] 319 -- 9A Elementary Matrices 319 -- 9B Commutators and K[subscript 1](R) 322 -- 9C Determinants 328 -- 9D The Bass K[subscript 1] of a Category 333 -- Chapter 10 Stability for K[subscript 1](R) 342 -- 10A Surjective Stability 343 -- 10B Injective Stability 348 -- Chapter 11 Relative K[subscript 1] 357 -- 11A Congruence Subgroups of GL[subscript n](R) 357 -- 11B Congruence Subgroups of SL[subscript n](R) 369 -- 11C Mennicke Symbols 374 -- Part IV Relations Among Matrices: K[subscript 2] 399 -- Chapter 12 K[subscript 2](R) and Steinberg Symbols 401 -- 12A Definition and Properties of K[subscript 2](R) 401 -- 12B Elements of St(R) and K[subscript 2](R) 413 -- Chapter 13 Exact Sequences 430 -- 13A The Relative Sequence 431 -- 13B Excision and the Mayer-Vietoris Sequence 456 -- 13C The Localization Sequence 481 -- Chapter 14 Universal Algebras 488 -- 14A Presentation of Algebras 489 -- 14B Graded Rings 493 -- 14C The Tensor Algebra 497 -- 14D Symmetric and Exterior Algebras 505 -- 14E The Milnor Ring 518 -- 14F Tame Symbols 534 -- 14G Norms on Milnor K-Theory 547 -- 14H Matsumoto's Theorem 557 -- Part V Sources of K[subscript 2] 567 -- Chapter 15 Symbols in Arithmetic 569 -- 15A Hilbert Symbols 569 -- 15B Metric Completion of Fields 572 -- 15C The p-Adic Numbers and Quadratic Reciprocity 580 -- 15D Local Fields and Norm Residue Symbols 595 -- Chapter 16 Brauer Groups 610 -- 16A The Brauer Group of a Field 610 -- 16B Splitting Fields 623 -- 16C Twisted Group Rings 629 -- 16D The K[subscript 2] Connection 636 -- A Sets, Classes, Functions 645 -- B Chain Conditions, Composition Series 647.

Print version record.

English.

eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - Worldwide