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| Preface | |
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| Rings and Modules | |
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| Preliminaries | |
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| Projective modules | |
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| Injective modules | |
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| Semi-simple rings | |
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| Hereditary rings | |
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| Semi-hereditary rings | |
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| Noetherian rings | |
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| Exercises | |
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| Additive Functors | |
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| Definitions | |
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| Examples | |
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| Operators | |
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| Preservation of exactness | |
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| Composite functors | |
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| Change of rings | |
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| Exercises | |
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| Satellites | |
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| Definition of satellites | |
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| Connecting homomorphisms | |
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| Half exact functors | |
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| Connected sequence of functors | |
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| Axiomatic description of satellites | |
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| Composite functors | |
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| Several variables | |
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| Exercises | |
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| Homology | |
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| Modules with differentiation | |
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| The ring of dual numbers | |
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| Graded modules, complexes | |
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| Double gradings and complexes | |
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| Functors of complexes | |
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| The homomorphism x | |
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| The homomorphism x (continuation) | |
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| Kunneth relations | |
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| Exercises | |
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| Derived Functors | |
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| Complexes over modules; resolutions | |
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| Resolutions of sequences | |
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| Definition of derived functors | |
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| Connecting homomorphisms | |
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| The functors ROT and LOT | |
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| Comparison with satellites | |
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| Computational devices | |
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| Partial derived functors | |
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| Sums, products, limits | |
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| The sequence of a map | |
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| Exercises | |
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| Derived Functors of 0 and Hom | |
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| The functors Tor and Ext | |
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| Dimension of modules and rings | |
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| Kunneth relations | |
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| Change of rings | |
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| Duality homomorphisms | |
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| Exercises | |
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| Integral Domains | |
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| Generalities | |
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| The field of quotients | |
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| Inversible ideals | |
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| Prufer rings | |
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| Dedekind rings | |
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| Abelian groups | |
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| A description of Tor1, (A,C) | |
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| Exercises | |
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| Augmented Rings | |
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| Homology and cohomology of an augmented ring | |
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| Examples | |
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| Change of rings | |
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| Dimension | |
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| Faithful systems | |
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| Applications to graded and local rings | |
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| Exercises | |
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| Associative Algebras | |
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| Algebras and their tensor products | |
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| Associativity formulae | |
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| The enveloping algebra Ae | |
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| Homology and cohomology of algebras | |
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| The Hochschild groups as functors of A | |
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| Standard complexes | |
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| Dimension | |
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| Exercises | |
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| Supplemented Algebras | |
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| Homology of supplemented algebras | |
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| Comparison with Hochschild groups | |
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| Augmented monoids | |
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| Groups | |
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| Examples of resolutions | |
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| The inverse process | |
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| Subalgebras and subgroups | |
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| Weakly injective and projective modules | |
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| Exercises | |
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| Products | |
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| External products | |
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| Formal properties of the products | |
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| Isomorphisms | |
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| Internal products | |
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| Computation of products | |
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| Products in the Hochschild theory | |
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| Products for supplemented algebras | |
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| Associativity formulae | |
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| Reduction theorems 225 Exercises | |
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| Finite Groups | |
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| Norms | |
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| The complete derived sequence | |
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| Complete resolutions | |
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| Products for finite groups | |
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| The uniqueness theorem | |
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| Duality | |
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| Examples | |
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| Relations with subgroups | |
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| Double cosets | |
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| p-groups and Sylow groups | |
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| Periodicity 260 Exercises | |
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| Lie Algebras | |
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| Lie algebras and their enveloping algebras | |
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| Homology and cohomology of Lie algebras | |
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| The Poincare-Witt theorem | |
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| Subalgebras and ideals | |
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| The diagonal map and its applications | |
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| A relation in the standard complex | |
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| The complex V(g) | |
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| Applications of the complex V(g) | |
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| Exercises | |
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| Extensions | |
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| Extensions of modules | |
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| Extensions of | |