An algebraic introduction to K-theory /
Magurn, Bruce A.,
An algebraic introduction to K-theory / Bruce A. Magurn. - Cambridge : Cambridge University Press, 2002. - 1 online resource (xiv, 676 pages) : digital, PDF file(s). - Encyclopedia of mathematics and its applications ; volume 87 . - Encyclopedia of mathematics and its applications ; v. 87. .
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
Groups of Modules: K[subscript 0] Free Modules Bases Matrix Representations Absence of Dimension Projective Modules Direct Summands Summands of Free Modules Grothendieck Groups Semigroups of Isomorphism Classes Semigroups to Groups Grothendieck Groups Resolutions Stability for Projective Modules Adding Copies of R Stably Free Modules When Stably Free Modules Are Free Stable Rank Dimensions of a Ring Multiplying Modules Semirings Burnside Rings Tensor Products of Modules Change of Rings K[subscript 0] of Related Rings G[subscript 0] of Related Rings K[subscript 0] as a Functor The Jacobson Radical Localization Sources of K[subscript 0] Number Theory Algebraic Integers Dedekind Domains Ideal Class Groups Extensions and Norms K[subscript 0] and G[subscript 0] of Dedekind Domains Group Representation Theory Linear Representations Representing Finite Groups Over Fields Semisimple Rings Characters Groups of Matrices: K[subscript 1] Definition of K[subscript 1] Elementary Matrices Commutators and K[subscript 1](R) Determinants The Bass K[subscript 1] of a Category Stability for K[subscript 1](R) Surjective Stability Injective Stability Relative K[subscript 1] Congruence Subgroups of GL[subscript n](R) Congruence Subgroups of SL[subscript n](R) Mennicke Symbols Relations Among Matrices: K[subscript 2] K[subscript 2](R) and Steinberg Symbols Definition and Properties of K[subscript 2](R) Elements of St(R) and K[subscript 2](R) Exact Sequences The Relative Sequence Excision and the Mayer-Vietoris Sequence The Localization Sequence Universal Algebras Presentation of Algebras Graded Rings The Tensor Algebra Symmetric and Exterior Algebras The Milnor Ring Tame Symbols Norms on Milnor K-Theory Matsumoto's Theorem Sources of K[subscript 2] Symbols in Arithmetic Hilbert Symbols Metric Completion of Fields The p-Adic Numbers and Quadratic Reciprocity Local Fields and Norm Residue Symbols Brauer Groups The Brauer Group of a Field Splitting Fields Twisted Group Rings The K[subscript 2] Connection A Sets, Classes, Functions Chain Conditions, Composition Series Part I 15 -- Chapter 1 17 -- 1A 17 -- 1B 26 -- 1C 38 -- Chapter 2 43 -- 2A 43 -- 2B 51 -- Chapter 3 57 -- 3A 57 -- 3B 71 -- 3C 83 -- 3D 95 -- Chapter 4 104 -- 4A 104 -- 4B 108 -- 4C 113 -- 4D 120 -- 4E 128 -- Chapter 5 133 -- 5A 133 -- 5B 135 -- 5C 141 -- Chapter 6 160 -- 6A 160 -- 6B 169 -- 6C 174 -- 6D 178 -- 6E 185 -- Part II 203 -- Chapter 7 205 -- 7A 205 -- 7B 212 -- 7C 224 -- 7D 230 -- 7E 242 -- Chapter 8 252 -- 8A 252 -- 8B 265 -- 8C 277 -- 8D 300 -- Part III 317 -- Chapter 9 319 -- 9A 319 -- 9B 322 -- 9C 328 -- 9D 333 -- Chapter 10 342 -- 10A 343 -- 10B 348 -- Chapter 11 357 -- 11A 357 -- 11B 369 -- 11C 374 -- Part IV 399 -- Chapter 12 401 -- 12A 401 -- 12B 413 -- Chapter 13 430 -- 13A 431 -- 13B 456 -- 13C 481 -- Chapter 14 488 -- 14A 489 -- 14B 493 -- 14C 497 -- 14D 505 -- 14E 518 -- 14F 534 -- 14G 547 -- 14H 557 -- Part V 567 -- Chapter 15 569 -- 15A 569 -- 15B 572 -- 15C 580 -- 15D 595 -- Chapter 16 610 -- 16A 610 -- 16B 623 -- 16C 629 -- 16D 636 -- 645 -- B 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.
9781107326002 (ebook)
K-theory.
QA612.33 / .M34 2002
512/.55
An algebraic introduction to K-theory / Bruce A. Magurn. - Cambridge : Cambridge University Press, 2002. - 1 online resource (xiv, 676 pages) : digital, PDF file(s). - Encyclopedia of mathematics and its applications ; volume 87 . - Encyclopedia of mathematics and its applications ; v. 87. .
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
Groups of Modules: K[subscript 0] Free Modules Bases Matrix Representations Absence of Dimension Projective Modules Direct Summands Summands of Free Modules Grothendieck Groups Semigroups of Isomorphism Classes Semigroups to Groups Grothendieck Groups Resolutions Stability for Projective Modules Adding Copies of R Stably Free Modules When Stably Free Modules Are Free Stable Rank Dimensions of a Ring Multiplying Modules Semirings Burnside Rings Tensor Products of Modules Change of Rings K[subscript 0] of Related Rings G[subscript 0] of Related Rings K[subscript 0] as a Functor The Jacobson Radical Localization Sources of K[subscript 0] Number Theory Algebraic Integers Dedekind Domains Ideal Class Groups Extensions and Norms K[subscript 0] and G[subscript 0] of Dedekind Domains Group Representation Theory Linear Representations Representing Finite Groups Over Fields Semisimple Rings Characters Groups of Matrices: K[subscript 1] Definition of K[subscript 1] Elementary Matrices Commutators and K[subscript 1](R) Determinants The Bass K[subscript 1] of a Category Stability for K[subscript 1](R) Surjective Stability Injective Stability Relative K[subscript 1] Congruence Subgroups of GL[subscript n](R) Congruence Subgroups of SL[subscript n](R) Mennicke Symbols Relations Among Matrices: K[subscript 2] K[subscript 2](R) and Steinberg Symbols Definition and Properties of K[subscript 2](R) Elements of St(R) and K[subscript 2](R) Exact Sequences The Relative Sequence Excision and the Mayer-Vietoris Sequence The Localization Sequence Universal Algebras Presentation of Algebras Graded Rings The Tensor Algebra Symmetric and Exterior Algebras The Milnor Ring Tame Symbols Norms on Milnor K-Theory Matsumoto's Theorem Sources of K[subscript 2] Symbols in Arithmetic Hilbert Symbols Metric Completion of Fields The p-Adic Numbers and Quadratic Reciprocity Local Fields and Norm Residue Symbols Brauer Groups The Brauer Group of a Field Splitting Fields Twisted Group Rings The K[subscript 2] Connection A Sets, Classes, Functions Chain Conditions, Composition Series Part I 15 -- Chapter 1 17 -- 1A 17 -- 1B 26 -- 1C 38 -- Chapter 2 43 -- 2A 43 -- 2B 51 -- Chapter 3 57 -- 3A 57 -- 3B 71 -- 3C 83 -- 3D 95 -- Chapter 4 104 -- 4A 104 -- 4B 108 -- 4C 113 -- 4D 120 -- 4E 128 -- Chapter 5 133 -- 5A 133 -- 5B 135 -- 5C 141 -- Chapter 6 160 -- 6A 160 -- 6B 169 -- 6C 174 -- 6D 178 -- 6E 185 -- Part II 203 -- Chapter 7 205 -- 7A 205 -- 7B 212 -- 7C 224 -- 7D 230 -- 7E 242 -- Chapter 8 252 -- 8A 252 -- 8B 265 -- 8C 277 -- 8D 300 -- Part III 317 -- Chapter 9 319 -- 9A 319 -- 9B 322 -- 9C 328 -- 9D 333 -- Chapter 10 342 -- 10A 343 -- 10B 348 -- Chapter 11 357 -- 11A 357 -- 11B 369 -- 11C 374 -- Part IV 399 -- Chapter 12 401 -- 12A 401 -- 12B 413 -- Chapter 13 430 -- 13A 431 -- 13B 456 -- 13C 481 -- Chapter 14 488 -- 14A 489 -- 14B 493 -- 14C 497 -- 14D 505 -- 14E 518 -- 14F 534 -- 14G 547 -- 14H 557 -- Part V 567 -- Chapter 15 569 -- 15A 569 -- 15B 572 -- 15C 580 -- 15D 595 -- Chapter 16 610 -- 16A 610 -- 16B 623 -- 16C 629 -- 16D 636 -- 645 -- B 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.
9781107326002 (ebook)
K-theory.
QA612.33 / .M34 2002
512/.55