Homological Algebra

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.

Henri Cartan was Professor of Mathematics at the University of Paris.

Preface	p. v
Rings and Modules	p. 3
Preliminaries	p. 3
Projective modules	p. 6
Injective modules	p. 8
Semi-simple rings	p. 11
Hereditary rings	p. 12
Semi-hereditary rings	p. 14
Noetherian rings	p. 15
Exercises	p. 16
Additive Functors	p. 18
Definitions	p. 18
Examples	p. 20
Operators	p. 22
Preservation of exactness	p. 23
Composite functors	p. 27
Change of rings	p. 28
Exercises	p. 31
Satellites	p. 33
Definition of satellites	p. 33
Connecting homomorphisms	p. 37
Half exact functors	p. 39
Connected sequence of functors	p. 43
Axiomatic description of satellites	p. 45
Composite functors	p. 48
Several variables	p. 49
Exercises	p. 51
Homology	p. 53
Modules with differentiation	p. 53
The ring of dual numbers	p. 56
Graded modules, complexes	p. 58
Double gradings and complexes	p. 60
Functors of complexes	p. 62
The homomorphism x	p. 64
The homomorphism x (continuation)	p. 66
Kunneth relations	p. 71
Exercises	p. 72
Derived Functors	p. 75
Complexes over modules; resolutions	p. 75
Resolutions of sequences	p. 78
Definition of derived functors	p. 82
Connecting homomorphisms	p. 84
The functors ROT and LOT	p. 89
Comparison with satellites	p. 90
Computational devices	p. 91
Partial derived functors	p. 94
Sums, products, limits	p. 97
The sequence of a map	p. 101
Exercises	p. 104
Derived Functors of 0 and Hom	p. 106
The functors Tor and Ext	p. 106
Dimension of modules and rings	p. 109
Kunneth relations	p. 112
Change of rings	p. 116
Duality homomorphisms	p. 119
Exercises	p. 122
Integral Domains	p. 127
Generalities	p. 127
The field of quotients	p. 129
Inversible ideals	p. 132
Prufer rings	p. 133
Dedekind rings	p. 134
Abelian groups	p. 135
A description of Tor1, (A,C)	p. 137
Exercises	p. 139
Augmented Rings	p. 143
Homology and cohomology of an augmented ring	p. 143
Examples	p. 146
Change of rings	p. 149
Dimension	p. 150
Faithful systems	p. 154
Applications to graded and local rings	p. 156
Exercises	p. 158
Associative Algebras	p. 162
Algebras and their tensor products	p. 162
Associativity formulae	p. 165
The enveloping algebra Ae	p. 167
Homology and cohomology of algebras	p. 169
The Hochschild groups as functors of A	p. 171
Standard complexes	p. 174
Dimension	p. 176
Exercises	p. 180
Supplemented Algebras	p. 182
Homology of supplemented algebras	p. 182
Comparison with Hochschild groups	p. 185
Augmented monoids	p. 187
Groups	p. 189
Examples of resolutions	p. 192
The inverse process	p. 193
Subalgebras and subgroups	p. 196
Weakly injective and projective modules	p. 197
Exercises	p. 201
Products	p. 202
External products	p. 202
Formal properties of the products	p. 206
Isomorphisms	p. 209
Internal products	p. 211
Computation of products
Products in the Hochschild theory	p. 216
Products for supplemented algebras	p. 219
Associativity formulae	p. 222
Reduction theorems 225 Exercises	p. 228
Finite Groups	p. 232
Norms	p. 232
The complete derived sequence	p. 235
Complete resolutions	p. 237
Products for finite groups	p. 242
The uniqueness theorem	p. 244
Duality	p. 247
Examples	p. 250
Relations with subgroups	p. 254
Double cosets	p. 256
p-groups and Sylow groups	p. 258
Periodicity 260 Exercises	p. 263
Lie Algebras	p. 266
Lie algebras and their enveloping algebras	p. 266
Homology and cohomology of Lie algebras	p. 270
The Poincare-Witt theorem	p. 271
Subalgebras and ideals	p. 274
The diagonal map and its applications	p. 275
A relation in the standard complex	p. 277
The complex V(g)	p. 279
Applications of the complex V(g)	p. 282
Exercises	p. 284
Extensions	p. 289
Extensions of modules	p. 289
Extensions of
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