Introduction to algebraic K-theory

Contents......Page 3
Preface......Page 5
Index of notation......Page 7
1 Modules......Page 11
1.1 Direct sums......Page 13
1.3 Projective modules......Page 15
2 The Grothendieck Group, K_0R......Page 18
2.1 Bimodules and tensor products......Page 22
2.2 Kq of a commutative ring......Page 25
2.3 The map K_0f : K_0R --> K_0S......Page 26
2.4 Products of rings......Page 29
2.5 Matrix rings......Page 33
2.6 The Jacobson radical, Wedderburn-Artin theory, and K_0 of semi-local rings......Page 35
2.7 The reduced group K_0R......Page 38
3.1 Rings of fractions......Page 40
3.2 Modules of fractions......Page 41
3.3 Localization......Page 43
3.4 Dual modules......Page 46
3.5 Exterior powers......Page 53
3.6 The prime spectrum and Zariski topology......Page 59
3.7 Connectedness in Spec R......Page 62
3.8 The determinant map det : K_0R --> Pic R......Page 67
4 K0 of Integral Domains and Dedekind Domains......Page 71
4.1 Examples: the calculation of K_0Z[\sqrt{-d}] for small d......Page 81
5 Categories......Page 88
5.1 Categories with product......Page 93
5.2 Categories with product and composition......Page 102
6 Whitehead Groups......Page 107
6.1 K_1C as a direct limit......Page 112
6.2 K_1 of a ring......Page 117
6.3 Determinants and SK_1......Page 122
6.4 The non-stable K_1 of a commutative ring......Page 126
6.5 Dieudonne determinants, and K_1 of semi-local rings......Page 132
7 Exact Sequences......Page 151
7.1 Cartesian squares, and the Mayer-Vietoris sequence......Page 164
8 Steinberg Groups and K_2R......Page 193
8.1 Generators for K_2R: the symbols {\alpha, \beta}......Page 203
8.2 Generators for K_2R: the symbols ......Page 218
8.3 The calculation of K_2(Z/nZ) and SK_1(Z, nZ)......Page 234
8.4 The non-triviality of K_2R......Page 239
8.5 Generators for K_2 of a semi-local ring......Page 242
Bibliography......Page 260
Index......Page 261