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Introduction 1

Ⅰ．Modules 10

1．Modules 11

2．The Group of Homomorphisms 16

3．Sums and Products 18

4．Free and Projective Modules 22

5．Projective Modules over a Principal Ideal Domain 26

6．Dualization,Injective Modules 28

7．Injective Modules over a Principal Ideal Domain 31

8．Cofree Modules 34

9．Essential Extensions 36

Ⅱ．Categories and Functors 40

1．Categories 40

2．Functors 44

3．Duality 48

4．Natural Transformations 50

5．Products and Coproducts：Universal Constructions 54

6．Universal Constructions（Continued）；Pull-backs and Push-outs 59

7．Adjoint Functors 63

8．Adjoint Functors and Universal Constructions 69

9．Abelian Categories 74

10．Projective,Injective,and Free Objects 81

Ⅲ．Extensions of Modules 84

1．Extensions 84

2．The Functor Ext 89

3．Ext Using Injectives 94

4．Computation of some Ext-Groups 97

5．Two Exact Sequences 99

6．A Theorem of Stein-Serre for Abelian Groups 106

7．The Tensor Product 109

8．The Functor Tor 112

Ⅳ．Derived Functors 116

1．Complexes 117

2．The Long Exact（Co）Homology Sequence 121

3．Homotopy 124

4．Resolutions 126

5．Derived Functors 130

6．The Two Long Exact Sequences of Derived Functors 136

7．The Functors Extn ∧ Using Projectives 139

8．The Functors？Using Injectives 143

9．Extn and n-Extensions 148

10．Another Characterization of Derived Functors 156

11．The Functor Tor∧ n 160

12．Change of Rings 162

Ⅴ．The Künneth Formula 166

1．Double Complexes 167

2．The Künneth Theorem 172

3．The Dual Künneth Theorem 177

4．Applications of the Künneth Formulas 180

Ⅵ．Cohomology of Groups 184

1．The Group Ring 186

2．Definition of（Co）Homology 188

3．H0,H0 191

4．H1,H1 with Trivial Coefficient Modules 192

5．The Augmentation Ideal,Derivations,and the Semi-Direct Product 194

6．A Short Exact Sequence 197

7．The（Co）Homology of Finite Cyclic Groups 200

8．The 5-Term Exact Sequences 202

9．H2,Hopf's Formula,and the Lower Central Series 204

10．H2 and Extensions 206

11．Relative Projectives and Relative Injectives 210

12．Reduction Theorems 213

13．Resolutions 214

14．The（Co）Homology of a Coproduct 219

15．The Universal Coefficient Theorem and the（Co）Homology of a Product 221

16．Groups and Subgroups 223

Ⅶ．Cohomology of Lie Algebras 229

1．Lie Algebras and their Universal Enveloping Algebra 229

2．Definition of Cohomology；H0,H1 234

3．H2 and Extensions 237

4．A Resolution of the Ground Field K 239

5．Semi-simple Lie Algebras 244

6．The two Whitehead Lemmas 247

7．Appendix：Hilbert's Chain-of-Syzygies Theorem 251

Ⅷ．Exact Couples and Spectral Sequences 255

1．Exact Couples and Spectral Sequences 256

2．Filtered Differential Objects 261

3．Finite Convergence Conditions for Filtered Chain Complexes 265

4．The Ladder of an Exact Couple 269

5．Limits 276

6．Rees Systems and Filtered Complexes 281

7．The Limit of a Rees System 288

8．Completions of Filtrations 291

9．The Grothendieck Spectral Sequence 297

Ⅸ．Satellites and Homology 306

1．Projective Classes of Epimorphisms 307

2．E-Derived Functors 309

3．E-Satellites 312

4．The Adjoint Theorem and Examples 318

5．Kan Extensions and Homology 320

6．Applications：Homology of Small Categories,Spectral Sequences 327

Ⅹ．Some Applications and Recent Developments 331

1．Homological Algebra and Algebraic Topology 331

2．Nilpotent Groups 335

3．Finiteness Conditions on Groups 339

4．Modular Representation Theory 344

5．Stable and Derived Categories 349

Bibliography 357

Index 359