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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 15Exercises 16 Chapter II. Additive Functors 18 1. Definitions 18 2.Examples 20 3. Operators 22 4. Preservation of exactness 23 5. Compositefunctors 27 6. Change of rings 28 Exercises 31 Chapter III. Satellites 331. Definition of satellites 33 2. Connecting homomorphisms 37 3. Half exactfunctors 39 4. Connected sequence of functors 43 5. Axiomatic descriptionof satellites 45 6. Composite functors 48 7. Several variables 49Exercises 51 Chapter IV. Homology 53 1. Modules with differentiation 53 2.The ring of dual numbers 56 3. Graded modules, complexes 58 4. Doublegradings and complexes 60 5. Functors of complexes 62 6. The homomorphism x64 7. The homomorphism x (continuation) 66 8. Kunneth relations 71Exercises 72 Chapter V. Derived Functors 75 1.Complexes over modules;resolutions 75 2.Resolutions of sequences 78 3.Definition of derivedfunctors 82 4.Connecting homomorphisms 84 5.The functors ROT and LOT 896.Comparison with satellites 90 7.Computational devices 91 8.Partialderived functors 94 9.Sums, products, limits 97 10.The sequence of a map101 Exercises 104 Chapter VI. Derived Functors of 0 and Hom 106 1. Thefunctors Tor and Ext 106 2. Dimension of modules and rings 109 3. Kunnethrelations 112 4. Change of rings 116 5. Duality homomorphisms 119Exercises 122 Chapter VII. Integral Domains 127 1. Generalities 127 2. Thefield 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 andcohomology of an augmented ring 143 2. Examples 146 3. Change of rings 1494. Dimension 150 5. Faithful systems 154 6. Applications to graded andlocal rings 156 Exercises 158 Chapter IX. Associative Algebras 162 1.Algebras and their tensor products 162 2. Associativity formulae 165 3. Theenveloping algebra Ae 167 4. Homology and cohomology of algebras 169 5. TheHochschild 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 groups185 3. Augmented monoids 187 4. Groups 189 5. Examples of resolutions 1926. The inverse process193 7. Subalgebras and subgroups 196 8. Weaklyinjective and projective modules 197 Exercises 201 Chapter XI. Products 2021. 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 algebras219 8. Associativity formulae 222 9. Reduction theorems 225 Exercises 228Chapter XII. Finite Groups 232 1. Norms 232 2. The complete derivedsequence 235 3. Complete resolutions 237 4. Products for finite groups 2425. The uniqueness theorem 244 6. Duality 247 7. Examples 250 8. Relationswith subgroups 254 9. Double cosets 256 10 p-groups and Sylow groups 25811. Periodicity 260 Exercises 263 Chapter XIII. Lie Algebras 266 1. Liealgebras and their enveloping algebras 266 2. Homology and cohomology of Liealgebras 270 3. The Poincare-Witt theorem 271 4. Subalgebras and ideals 2745. The diagonal map and its applications 275 6. A relation in the standardcomplex 277 7. The complex V(g) 279 8. Applications of the complex V(g) 282Exercises 284 Chapter XIV. Extensions 289 1. Extensions of modules 289 2.Extensions of associative algebras 293 3. Extensions of supplementedalgebras 295 4. Extensions of groups 299 5. Extensions of Lie algebras 304Exercises 308 Chapter XV. Spectral Sequences 315 1. Filtrations andspectral 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 336Chapter XVI. Applications of Spectral Sequences 340 1. Partial derivedfunctors 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 almostzero theory 358 Exercises 360 Chapter XVII. Hyperhomology 362 1.Resolutions of complexes 362 2. The invariants 366 3. Regularity conditions368 4. Mapping theorems 371 5. Kunneth relations 372 6. Balanced functors374 7. Composite functors 376 Appendix: Exact categories, by David A.Buchsbaum 379 List of Symbols 387 Index of Terminology 389