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Homological Algebra (PMS-19), Volume 19 : Princeton Landmarks in Mathematics - Henry Cartan

Homological Algebra (PMS-19), Volume 19

Princeton Landmarks in Mathematics

Paperback Published: 19th December 1999
ISBN: 9780691049915
Number Of Pages: 408

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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.

Prefacep. v
Rings and Modulesp. 3
Preliminariesp. 3
Projective modulesp. 6
Injective modulesp. 8
Semi-simple ringsp. 11
Hereditary ringsp. 12
Semi-hereditary ringsp. 14
Noetherian ringsp. 15
Exercisesp. 16
Additive Functorsp. 18
Definitionsp. 18
Examplesp. 20
Operatorsp. 22
Preservation of exactnessp. 23
Composite functorsp. 27
Change of ringsp. 28
Exercisesp. 31
Satellitesp. 33
Definition of satellitesp. 33
Connecting homomorphismsp. 37
Half exact functorsp. 39
Connected sequence of functorsp. 43
Axiomatic description of satellitesp. 45
Composite functorsp. 48
Several variablesp. 49
Exercisesp. 51
Homologyp. 53
Modules with differentiationp. 53
The ring of dual numbersp. 56
Graded modules, complexesp. 58
Double gradings and complexesp. 60
Functors of complexesp. 62
The homomorphism xp. 64
The homomorphism x (continuation)p. 66
Kunneth relationsp. 71
Exercisesp. 72
Derived Functorsp. 75
Complexes over modules; resolutionsp. 75
Resolutions of sequencesp. 78
Definition of derived functorsp. 82
Connecting homomorphismsp. 84
The functors ROT and LOTp. 89
Comparison with satellitesp. 90
Computational devicesp. 91
Partial derived functorsp. 94
Sums, products, limitsp. 97
The sequence of a mapp. 101
Exercisesp. 104
Derived Functors of 0 and Homp. 106
The functors Tor and Extp. 106
Dimension of modules and ringsp. 109
Kunneth relationsp. 112
Change of ringsp. 116
Duality homomorphismsp. 119
Exercisesp. 122
Integral Domainsp. 127
Generalitiesp. 127
The field of quotientsp. 129
Inversible idealsp. 132
Prufer ringsp. 133
Dedekind ringsp. 134
Abelian groupsp. 135
A description of Tor1, (A,C)p. 137
Exercisesp. 139
Augmented Ringsp. 143
Homology and cohomology of an augmented ringp. 143
Examplesp. 146
Change of ringsp. 149
Dimensionp. 150
Faithful systemsp. 154
Applications to graded and local ringsp. 156
Exercisesp. 158
Associative Algebrasp. 162
Algebras and their tensor productsp. 162
Associativity formulaep. 165
The enveloping algebra Aep. 167
Homology and cohomology of algebrasp. 169
The Hochschild groups as functors of Ap. 171
Standard complexesp. 174
Dimensionp. 176
Exercisesp. 180
Supplemented Algebrasp. 182
Homology of supplemented algebrasp. 182
Comparison with Hochschild groupsp. 185
Augmented monoidsp. 187
Groupsp. 189
Examples of resolutionsp. 192
The inverse processp. 193
Subalgebras and subgroupsp. 196
Weakly injective and projective modulesp. 197
Exercisesp. 201
Productsp. 202
External productsp. 202
Formal properties of the productsp. 206
Isomorphismsp. 209
Internal productsp. 211
Computation of products
Products in the Hochschild theoryp. 216
Products for supplemented algebrasp. 219
Associativity formulaep. 222
Reduction theorems 225 Exercisesp. 228
Finite Groupsp. 232
Normsp. 232
The complete derived sequencep. 235
Complete resolutionsp. 237
Products for finite groupsp. 242
The uniqueness theoremp. 244
Dualityp. 247
Examplesp. 250
Relations with subgroupsp. 254
Double cosetsp. 256
p-groups and Sylow groupsp. 258
Periodicity 260 Exercisesp. 263
Lie Algebrasp. 266
Lie algebras and their enveloping algebrasp. 266
Homology and cohomology of Lie algebrasp. 270
The Poincare-Witt theoremp. 271
Subalgebras and idealsp. 274
The diagonal map and its applicationsp. 275
A relation in the standard complexp. 277
The complex V(g)p. 279
Applications of the complex V(g)p. 282
Exercisesp. 284
Extensionsp. 289
Extensions of modulesp. 289
Extensions of
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691049915
ISBN-10: 0691049912
Series: Princeton Landmarks in Mathematics
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 408
Published: 19th December 1999
Country of Publication: US
Dimensions (cm): 24.13 x 15.88  x 3.18
Weight (kg): 0.57

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