TY  - BOOK
AU  - Magurn,Bruce A.
TI  - An algebraic introduction to K-theory
T2  - Encyclopedia of mathematics and its applications
SN  - 9781107326002 (ebook)
AV  - QA612.33 .M34 2002
U1  - 512/.55 21
PY  - 2002///
CY  - Cambridge
PB  - Cambridge University Press
KW  - K-theory
N1  - 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
N2  - 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
UR  - https://doi.org/10.1017/CBO9781107326002
ER  -