In 1942 came the first move forward towards homological algebra as we know it today, with the arrival of a paper by Samuel Eilenberg and Saunders MacLane. In it we find Hom and Ext defined for the very first time, and along with it the notions of a functor and natural isomorphism. These were needed to provide a precise language for talking about the properties of $Hom(A,B)$; in particular the fact that it varies naturally, contravariantly in $A$ and covariantly in $B$.

Cartan and Eilenberg’s book was truly a revolution in the subject, and in fact it was here that the term “Homological Algebra” was first coined. The book used derived functors in a systematic way which united all the previous homology theories, which in the past ten years had arisen in group theory, Lie algebras and algebraic geometry. The sheer list of terms that were first defined in this book may give the reader an idea of how much of this project is due to the existence of that one book! They defined what it means for an object to be projective or injective, and defined the notions of projective and injective resolutions. It is here that we find the first mention of $Hom$ being left exact and the first occurrence of $Ext^n$ as the right derived functors of $Hom$.

Contents

Preface

Chapter I. Rings and Modules 3

1. Preliminaries 3

2. Projective modules 6

3. Injective modules 8

4. Semi-simple rings 11

5. Hereditary rings 12

6. Semi-hereditary rings 14

7. Noetherian rings 15

Exercises 16

Chapter II. Additive Functors 18

1. Definitions 18

2. Examples 20

3. Operators 22

4. Preservation of exactness 23

5. Composite functors 27

6. Change of rings 28

Exercises 31

Chapter III. Satellites 33

1. Definition of satellites 33

2. Connecting homomorphisms 37

3. Half exact functors 39

4. Connected sequence of functors 43

5. Axiomatic description of satellites 45

6. Composite functors 48

7. Several variables 49

Exercises 51

Chapter IV. Homology 53

1. Modules with differentiation 53

2. The ring of dual numbers 56

3. Graded modules, complexes 58

4. Double gradings and complexes 60

5. Functors of complexes 62

6. The homomorphism a 64

7. The homomorphism a (continuation) 66

8. Künneth relations 71

Exercises 72

Chapter V. Derived Functors 75

1. Complexes over modules; resolutions 75

2. Resolutions of sequences 78

3. Definition of “derived functors 82

4. Connecting homomorphisms 84

5. The functors R?T and L^T 89

6. Comparison with satellites 90

7. Computational devices 91

8. Partial derived functors 94

9. Sums, products, limits 97

10. The sequence of a map 101

Exercises 104

Chapter VI. Derived Functors of ⊗ and Hom 106

1. The functors Tor and Ext 106

2. Dimension of modules and rings 109

3. Künneth relations 112

4. Change of rings 116

5. Duality homomorphisms 119

Exercises 122

Chapter VII. Integral Domains 127

1. Generalities 127

2. The field of quotients 129

3. Inversible ideals 132

4. Prufer rings 133

5. Dedekind rings 134

6. Abelian groups 135

7. A description of Torj (A,C) 137

Exercises 139

Chapter VIII. Augmented Rings 143

1. Homology and cohomology of an augmented ring 143

2. Examples 146

3. Change of rings 149

4. Dimension 150

5. Faithful systems 154

6. Applications to graded and local rings 156

Exercises 158

Chapter IX. Associative Algebras 162

1. Algebras and their tensor products 162

2. Associativity formulae 165

3. The enveloping algebra A* 167

4. Homology and cohomology of algebras 169

5. The Hochschild groups as functors of A 171

6. Standard complexes 174

7. Dimension 176

Exercises 180

Chapter X. Supplemented Algebras 182

1. Homology of supplemented algebras 182

2. Comparison with Hochschild groups 185

3. Augmented monoids 187

4. Groups 189

5. Examples of resolutions 192

6. The inverse process 193

7. Subalgebras and subgroups 196

8. Weakly injective and projective modules 197

Exercises 201

Chapter XI. Products 202

1. External products 202

2. Formal properties of the products 206

3. Isomorphisms 209

4. Internal products 211

5. Computation of products 213

6. Products in the Hochschild theory 216

7. Products for supplemented algebras 219

8. Associativity formulae 222

9. Reduction theorems 225

Exercises 228

Chapter XII. Finite Groups 232

1. Norms 232

2. The complete derived sequence 235

3. Complete resolutions 237

4. Products for finite groups 242

5. The uniqueness theorem 244

6. Duality 247

7. Examples 250

8. Relations with subgroups 254

9. Double cosets 256

10. p-groups and Sylow groups 258

11. Periodicity 260

Exercises 263

Chapter XIII. Lie Algebras 266

1. Lie algebras and their enveloping algebras 266

2. Homology and cohomology of Lie algebras 270

3. The Poincare-Witt theorem 271

4. Subalgebras and ideals 274

5. The diagonal map and its applications 275

6. A relation in the standard complex 277

7. The complex F(g) 279

8. Applications of the complex K(g) 282

Exercises 284

Chapter XIV. Extensions 289

1. Extensions of modules 289

2. Extensions of associative algebras 293

3. Extensions of supplemented algebras 295

4. Extensions of groups 299

5. Extensions of Lie algebras 304

Exercises 308

Chapter XV. Spectral Sequences 315

1. Filtrations and spectral sequences 315

2. Convergence 319

3. Maps and homotopies 321

4. The graded case 323

5. Induced homomorphisms and exact sequences 325

6. Application to double complexes 330

7. A generalization 333

Exercises 336

Chapter XVI. Applications of Spectral Sequences 340

1. Partial derived functors 340

2. Functors of complexes 342

3. Composite functors 343

4. Associativity formulae 345

5. Applications to the change of rings 347

6. Normal subalgebras 349

7. Associativity formulae using diagonal maps 351

8. Complexes over algebras 352

9. Topological applications 355

10. The almost zero theory 358

Exercises 360

Chapter XVII. Hyperhomology 362

1. Resolutions of complexes 362

2. The invariants 366

3. Regularity conditions 368

4. Mapping theorems 371

5. Künneth relations 372

6. Balanced functors 374

7. Composite functors 376

Appendix: Exact categories, by David A. Buchsbaum 379