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Homological Algebra
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Preface v 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 x 64 7. The homomorphism x (continuation) 66 8. Kunneth 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 ROT and LOT 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 0 and Hom 106 1. The functors Tor and Ext 106 2. Dimension of modules and rings 109 3. Kunneth 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 Tor1, (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 Ae 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 process193 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 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 V(g) 279 8. Applications of the complex V(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. Kunneth relations 372 6. Balanced functors 374 7. Composite functors 376 Appendix: Exact categories, by David A. Buchsbaum 379 List of Symbols 387 Index of Terminology 389

Henri Cartan, formerly Professor of Mathematics at the University of Paris, is a Fellow of the Royal Society. Samuel Eilenberg (1914-1998) was Professor of Mathematics at Columbia University. Both were founding members of the Bourbaki and both received the Wolf Prize in Mathematics.