preface
chapter i. rings and modules
1. preliminaries
2. projective modules
3. injective modules
4. semi-simple rings
5. hereditary rings
6. semi-hereditary rings
7. noetherian tings
exercises
chapter ii. additive functors
1. definitions
2. examples
3. operators
4. preservation of exactness
5. composite functors
6. change of rings
exercises
chapter iii. satellites
1. definition of satellites
2. connecting homomorphisms
3. half exact functors
4. connected sequence of functors
5. axiomatic description of satellites
6. composite functors
7. several variables
exercises
chapter iv. homology
1. modules with differentiation
2. the ring of dual numbers
3. graded modules, complexes
4. double gradings and complexes
5. functors of complexes
6. the homomorphism
7. the homomorphism a continuation
8. kiinneth relations
exercises
chapter v. derived functors
1. complexes over modules; resolutions
2. resolutions of sequences
3. definition of derived functors
4. connecting homomorphisms
5. the functors rot and lot
6. comparison with satellites
7. computational devices
8. partial derived functors
9. sums, products, limits
i0. the sequence of a map
exercises
chapter vi. derived functors of ~ and hem
1. the functors tor and ext
2. dimension of modules and rings
3. kiinneth relations
4. change of rings
5. duality homomorphisms
exercises
chapter vli. integral domains
1. generalities
2. the field of quotients
3. inversible ideals
4. priifer rings
5. dedekind rings
6. abelian groups
7. a description of torx a,c
exercises
chapter viii. augmented rings
1. homology and cohomology of an augmented ring
2. examples
3. change of rings
4. dimension
5. faithful systems
6. applications to graded and local rings
exercises
chapter ix. associative algebras
1. algebras and their tensor products
2. associativity formulae
3. the enveloping algebra a~
4. homology and cohomology of algebras
5. the hochschild groups as functors of a
6. standard complexes
7. dimension
exercises
chapter x. supplemented algebras
1. homology of supplemented algebras
2. comparison with hochschild groups
3. augmented monoids
4. groups
5. examples of resolutions
6. the inverse process
7. subalgebras and subgroups
8. weakly injective and projective modules
exercises
chapter xi. products
1. external products
2. formal properties of the products
3. lsomorphisms
4. internal products
5. computation of products
6. products in the hochschild theory
7. products for supplemented algebras
8. associativity formulae
9. reduction theorems
exercises
chapter xii. finite groups
1. norms
2. the complete derived sequence
3. complete resolutions
4. products for finite groups
5. the uniqueness theorem
6. duality
7. examples
8. relations with subgroups
9. double cosets
10. p-groups and sylow groups
1. periodicity
exercises
chapter xlli. lie algebras
1. lie algebras and their enveloping algebras
2. homology and cohomology of lie algebras
3. the poincare-witt theorem
4. subaigebras and ideals
5. the diagonal map and its applications
6. a relation in the standard complex
7. the complex vg
8. applications of the complex vg
exercises
chapter xiv. extensions
1. extensions of modules
2. extensions of associative algebras
3. extensions of supplemented algebras
4. extensions of groups
5. extensions of lie algebras
exercises
chapter xv. spectral sequences
1. filtrations and spectral sequences
2. convergence
3. maps and homotopies
4. the graded case
5. induced homomorphisms and exact sequences
6. application to double complexes
7. a generalization
exercises
chapter xvi. applications of spectral sequences
1. partial derived functors
2. functors of complexes
3. composite functors
4. associativity formulae
5. applications to the change of rings
6. normal subalgebras
7. associativity formulae using diagonal maps
8. complexes over algebras
9. topological applications
10. the almost zero theory
exercises
chapter xvll. hyperhomology
1. resolutions of complexes
2. the invariants
3. regularity conditions
4. mapping theorems
5. kiinneth relations
6. balanced functors
7. composite functors
appendix: exact categories, by david a. buchsbaum
list of symbols
index of terminology
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