GTM 196 基本同调代数PDF电子书下载

1 Categories 1

2 Modules 11

2.1 Generalities 11

2.2 Tensor Products 14

2.3 Exactness of Functors 22

2.4 Projectives,Injectives,and Flats 28

3 Ext and Tor 39

3.1 Complexes and Projective Resolutions 39

3.2 Long Exact Sequences 47

3.3 Flat Resolutions and Injective Resolutions 55

3.4 Consequences 66

4 Dimension Theory 73

4.1 Dimension Shifting 73

4.2 When Flats are Projective 79

4.3 Dimension Zero 82

4.4 An Example 91

5 Change of Rings 99

5.1 Computational Considerations 99

5.2 Matrix Rings 104

5.3 Polynomials 106

5.4 Quotients and Localization 110

6 Derived Functors 123

6.1 Additive Functors 123

6.2 Derived Functors 126

6.3 Long Exact Sequences—Ⅰ．Existence 130

6.4 Long Exact Sequences—Ⅱ．Naturality 140

6.5 Long Exact Sequences—Ⅲ．Weirdness 147

6.6 Universality of Ext 151

7 Abstract Homological Algebra 165

7.1 Living Without Elements 165

7.2 Additive Categories 169

7.3 Kernels and Cokernels 173

7.4 Cheating with Projectives 186

7.5 （Interlude）Arrow Categories 202

7.6 Homology in Abelian Categories 213

7.7 Long Exact Sequences 225

7.8 An Alternative for Unbalanced Categories 239

8 Colimits and Tor 257

8.1 Limits and Colimits 257

8.2 Adjoint Functors 264

8.3 Directed Colimits,？,and Tor 270

8.4 Lazard's Theorem 274

8.5 Weak Dimension Revisited 280

9 Odds and Ends 285

9.1 Injective Envelopes 285

9.2 Universal Coefficients 290

9.3 The Künneth Theorems 296

9.4 Do Connecting Homomorphisms Commute？ 309

9.5 The Ext Product 318

9.6 The Jacobson Radical,Nakayama's Lemma,and Quasilocal Rings 324

9.7 Local Rings and Localization Revisited（Expository） 331

A GCDs,LCMs,PIDs,and UFDs 337

B The Ring of Entire Functions 345

C The Mitchell-Freyd Theorem and Cheating in Abelian Cat-egories 359

D Noether Correspondences in Abelian Categories 363

Solution Outlines 373

References 383

Symbol Index 389

Index 391