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Cartan E., Eilenberg S. — Homological Algebra, Vol. 19
Cartan E., Eilenberg S. — Homological Algebra, Vol. 19



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Название: Homological Algebra, Vol. 19

Авторы: Cartan E., Eilenberg S.

Аннотация:

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.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1956

Количество страниц: 408

Добавлена в каталог: 11.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Allowable family      154
Antipodism      222 269 351
Associative algebra      162ff
Associative algebra, cohomology of      169
Associative algebra, dimension of      176
Associative algebra, direct product of      172
Associative algebra, enveloping algebra of      167
Associative algebra, graded      164
Associative algebra, homology of      169
Associative algebra, homomorphism of      162
Associative algebra, normal homomorphism of      349
Associative algebra, normal subalgebra of      350
Associative algebra, normalized standard complex of      176
Associative algebra, projective, with respect to map      312
Associative algebra, standard complex of      175
Augmentation      75
Augmentation, epimorphism      143
Augmentation, ideal      143
Augmentation, idempotent,      221
Augmentation, module      143
Augmentation, unit      188
Augmentation, zero      188
Augmented ring      143
Augmented ring, cohomology of      143
Augmented ring, homology of      143
Augmented ring, map of      149
bi-graded module      60
bi-graded module, associated graded module of      60
bi-graded module, bihomogeneous element of      60
bi-graded module, bihomogeneous submodule of,      60
bi-graded module, homomorphism of      60
bi-graded module, negative homomorphism of      60
bi-graded module, positive homomorphism of      60
Characteristic element      227
COMPLEX      58
Complex, acyclic left      75
Complex, acyclic right      78
Complex, homomorphism of      59
Complex, injective resolution of      365
Complex, injective right      78
Complex, left      75
Complex, projective left      75
Complex, projective resolution of      363
Complex, right      78
Complex, split      70
Connected sequence of functors      43
Connected sequence of functors, homomorphism of multiply      87
Connected sequence of functors, map of      45
Connected sequence of functors, multiply      87
Connecting homomorphism      43 334
Contravariant $\varphi$-extension      29
Covariant $\varphi$-extension      29
Crossed homomorphism      168 270
Crossed homomorphism, principal      169 270
Crossed homomorphism, principal, with respect to map      312
Dedekind ring      134
Derived sequence of map      101
Diagonal map      211 351
Diagonal map, associative      212
Diagonal map, commutative      212
Diagonal map, commutative of a Lie algebra      275
DIMENSION      109 ff
Dimension, injective      111
Dimension, left global      111
Dimension, projective      109
Dimension, right global      111
Dimension, weak      122
Divisible element      127
Double complex      60
Double complex, first spectral sequence of      331
Double complex, homomorphism of,      61
Double complex, invariants of      331
Double complex, left      363
Double complex, right      365
Double complex, second spectral sequence of      331
Dual category      381
Duality theorem      249
Duality theorem, integral      250
Edge homomorphisms      330
Epimorphism      4
Exact category      379
Exact sequence      4
Exact sequence of terms of low degree      330
Exact sequence, normal      79
Exact sequence, normal form of      79
Exact sequence, split      5
Extension of ring of operators      163
Extensions      289 ff
Extensions, Baer multiplication of      290
Extensions, characteristic class of      290
Extensions, equivalent      289 293 299 304
Extensions, inessential      293 299 304
Extensions, split class of,      290
Faithful set      154
Filtration      315 ff
Filtration, compatible with differentiation      315
Filtration, complementary degree of      323
Filtration, convergent      321
Filtration, degree of,      323
Filtration, regular      324
Filtration, strongly convergent,      321
Filtration, total degree of      323
Filtration, weakly convergent      319
functor      18 ff
Functor, additive      19
Functor, contravariant      18
Functor, covariant      18
Functor, derived      83
Functor, derived sequence of a      102
Functor, exact      23
Functor, half exact      24
Functor, left balanced      97
Functor, left derived      84
Functor, left exact      24
Functor, partial derived      94
Functor, right balanced      96
Functor, right derived      83
Functor, right exact      24
Graded module      58
Graded module, homogeneous component of      58
Graded module, homogeneous submodule of      58
Graded module, homomorphism of      58
Graded module, negative      58
Graded module, positive      58
Graded ring      146
Graded ring, graded module over a      154
groups      189ff
Groups, complete derived sequence of a finite,      235
Groups, complete resolution for a finite,      240
Groups, cyclic      250
Groups, finite      232 ff
Groups, free      192
Groups, free abelian      193
Groups, generalized quaternion      253
Groups, projective, with respect to map      312
Hereditary ring      13
Homomorphism      3
Homomorphism, coimage of      3
Homomorphism, cokernel of      3
Homomorphism, compatible with filtration      321
Homomorphism, direct family of      4
Homomorphism, image of      3
Homomorphism, kernel of      3
Homotopy      54 59 61 72
Homotopy of order $\leqq k$      321
Homotopy, contracting,      214
Hypercohomology module      366
Hyperhomology module      368
Injective resolution      78
Injective resolution of a complex      365
Invariant elements      170 183
Inversible ideal      132
Isomorphism      4
Iterated connecting homomorphism      44
Lie algebra      266 ff
Lie algebra, abelian      268
Lie algebra, cohomology of      270
Lie algebra, constants of structure of      286
Lie algebra, enveloping algebra of      267
Lie algebra, homology of      270
Lie algebra, representation of      267
Local ring      147
Mapping theorem      150
Maximal generator      260
Module      3
Module with differentiation      53
Module, bimodule      22
Module, direct product of      4
Module, direct sum of      4
Module, divisible      127
Module, flat      122
Module, free      5
Module, injective      8
Module, map of with differentiation      54
Module, projective      6
Module, proper      154
Module, semi-simple      11
Module, simple,      11
Module, torsion      127
Module, torsion-free      127
Module, weakly injective      198 199
Module, weakly projective      197 199
Monoid      148 187ff
Monoid, augmented      188
Monoid, free      148 192
Monoid, free abelian      193
Monoid, ring of a      148
Monomorphism      4
Natural equivalence      20
Noetherian ring      15
Norm homomorphism      232
Opposite ring      109
p-primary component      258
Period      261
Products      202 ff
Products, cap-pairing      212
Products, cap-product      211
Products, cup-pairing      212
Products, cup-product      211
Products, modified      224 247
Products, modified of second kind      229
Projective resolution      75
Projective resolution, split form of      214
Proper operators      355
Pruefer ring      133
Reduction Theorem      227
Restriction homomorphism      254
Ring of dual numbers      56
Satellites      33 ff
Satellites, left      36
Satellites, right      36
Semi-hereditary ring      14
Spectral sequence      319
Stable element      257
Supplemented algebra      182ff
Supplemented algebra, cohomology of      182
Supplemented algebra, homology of      182
Supplemented algebra, homomorphism of      184
Tensor product      21
Tensor product, skew      164
Torsion element      127
Total differentiation operator      61
Transfer homomorphism      254
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