Homological Algebra (PMS-19), Volume 19

✍ Scribed by Henry Cartan; Samuel Eilenberg

• Publisher: Princeton University Press
• Year: 2016
• Tongue: English
• Leaves: 407
• Series: Princeton Mathematical Series; 28
• Category: Library

✦ Synopsis

When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried to unify the fields and to construct the framework of a fully fledged theory. The invasion of algebra had occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. This book presents a single homology (and also cohomology) theory that embodies all three; a large number of results is thus established in a general framework. Subsequently, each of the three theories is singled out by a suitable specialization, and its specific properties are studied.

The starting point is the notion of a module over a ring. The primary operations are the tensor product of two modules and the groups of all homomorphisms of one module into another. From these, "higher order" derived of operations are obtained, which enjoy all the properties usually attributed to homology theories. This leads in a natural way to the study of "functors" and of their "derived functors."

This mathematical masterpiece will appeal to all mathematicians working in algebraic topology.

✦ Table of Contents

Preface
Contents
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 rings
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 a
7. The homomorphism a (continuation)
8. Kunneth 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 R°T and L0T
6. Comparison with satellites
7. Computational devices
8. Partial derived functors
9. Sums, products, limits
10. The sequence of a map
Exercises
Chapter VI. Derived Functors of 0 and Horn
1. The functors Tor and Ext
2. Dimension of modules and rings
3. Kunneth relations
4. Change of rings
5. Duality homomorphisms
Exercises
Chapter VII. 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 e
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. Isomorphisms
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
11. Periodicity
Exercises
Chapter XIII. Lie Algebras
1. Lie algebras and their enveloping algebras
2. Homology and cohomology of Lie algebras
3. The Poincare-Witt theorem
4. Subalgebras and ideals
5. The diagonal map and its applications
6. A relation in the standard complex
7. The complex F(q)
8. Applications of the complex F(g)
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 XVII. Hyperhomology
1. Resolutions of complexes
2. The invariants
3. Regularity conditions
4. Mapping theorems
5. Kunneth relations
6. Balanced functors
7. Composite functors
Appendix: Exact categories
List of Symbols
Index of Terminology