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

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

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.

This is an introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain). The presentation is almost entirely self-contained, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. No experience with analysis, geometry, number theory or topology is assumed. Within the context of linear algebra, K-theory organises and clarifies the relations among ideal class groups, group representations, quadratic forms, dimensions of a ring, determinants, quadratic reciprocity and Brauer groups of fields. By including introductions to standard algebra topics (tensor products, localisation, Jacobson radical, chain conditions, Dedekind domains, semi-simple rings, exterior algebras), the author makes algebraic K-theory accessible to first-year graduate students and other mathematically sophisticated readers. Even if your algebra is rusty, you can read this book; the necessary background is here, with proofs.