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Magurn B.A. — An algebraic introduction to k-theory
Magurn B.A. — An algebraic introduction to k-theory

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Название: An algebraic introduction to k-theory

Автор: Magurn B.A.

Аннотация:

This book is both an introduction to K-theory and a text in algebra. These two roles are entirely compatible. On the one hand, nothing more than the basic algebra of groups, rings, and modules is needed to explain the clasical algebraic K-theory. On the other hand, K-theory is a natural organizing principle for the standard topics of a second course in algebra, and these topics are presented carefully here. The reader will not only learn algebraic K-theory, but also Dedekind domains, class groups, semisimple rings, character theory, quadratic forms, tensor products, localization, completion, tensor algebras, symmetric algebras, exterior algebras, central simple algebras, and Brauer groups. The presentation is self-contained, with all the necessary background and proofs, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. The prerequisites are minimal: just a first semester of algebra (including Galois theory and modules over a principal ideal domain). No experience with homological algebra, analysis, geometry, number theory, or topology is assumed. The author has successfuly used this text to teach algebra to first year graduate students. Selected topics can be used to construct a variety of one-semester courses; coverage of the entire text requires a full year.


Язык: en

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 676

Добавлена в каталог: 17.05.2011

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abelianization      322 323
Absolute value      572
Absolute value, Archimedean      573
Absolute value, trivial      573
Absolute value, ultrametric      573
action      135
Acts through bijections      188
Acts through injections      188
Additive abelian group      1
Algebra      2 489
Algebra Azumaya      618
Algebra, central      616
Algebra, cyclic      630—631
Algebra, division      285
Algebra, exterior      507
Algebra, finite-dimensional      613
Algebra, graded      494
Algebra, homomorphism      3 489
Algebra, power norm residue      631
Algebra, presentation      491
Algebra, separable      628
Algebra, sub-      489 613
Algebra, symmetric      505
Algebra, tensor      500
Algebraic integer      210
Anisotropic part      158
Anisotropic space      157
Annihilator      232
Anti-isomorphism of categories      9
Antihomomorphism      323
arrow      2
Arrow, identity      3
Arrow, inverse      6
Augmentation      276
Automorphism      6
Balanced, R-      142
Basis      17 74
Bass $K_{1}$ of a category      334
Bass — Milnor — Serre theorem      393
Bass — Whitehead group      325
Bimodule      144
Binary operation on a category      59
Binary operation, associative      59
Binary operation, commutative      59
Binary operation, identity of      59
Block      277 280
Block addition      34
Block multiplication      34—35
Boundedly generated      59
Brauer functor      623
Brauer Grothendieck group      619
Brauer group      619
Burnside ring      138
Cancellation, n-      105
Cancellation, Witt      80
Cartan map      92
Cartan matrix      93
Category      2
Category, concrete      4
Category, modest      58
Category, small      13
Center      186 288 615
Central extension      407
centralizer      300 622
CHARACTER      302
Character ring      303
Character table      305
Character, afforded by a module      302
Character, irreducible      302
Character, regular      303
Character, trivial      303
Character, virtual      304
Class function      302
Class number      226
Codomain      3
Colimit      438
Column, equivalent      321
Column, finite      38
Commutant      622
Commutator      322
Commutator subgroup      322
Commutator subgroup, mixed      358
Complement      48
Completion, group      71—72
Completion, metric      575
Composite arrow      2
Composite element      206
Conductor      437
Conductor, square      437
Cone      107 593
Congruence subgroup      358 369
Congruence subgroup, property      370
Congruent matrices      66
Countable cartesian product      86
Countable direct sum      86
Countably generated      60
Crossed product      630
Cyclotomic polynomial      429 473
Dedekind domain      214
Defining relations      22 25 401 491
Degree of a character      302
Degree of a representation      253
Degree of an Azumaya algebra      624
Derived series      327
Derived subgroups      327
Determinant      328 513 515
Determinant, Whitehead — Bass      330
Devissage      89—90
Dicyclic group      264
Direct limit      107
Direct sum      43 267
Direct sum of matrices      36
Direct sum, external      43
Direct sum, internal      43 267
Direct summand      48
Dirichlet's theorem      395
Discrete subgroup      579
Discriminant      158
Disjoint union      136
Divisible      525
Divisible, uniquely      525
Domain      3 40
Double centralizer property      300
Double of a ring      361 431
Dual basis      68
Dual functor      166
Dual linear map      67
Dual module      166
Efficiently related      96
Eilenberg swindle      113
Endomorphism      2
Erase      35
Euler characteristic      100
Even maximal ideal      597
Exact sequence      44
Exact sequence, short      44
Exact sequence, split      47
Excision      456
Excision map      457
Extended ideal      230
Extension homomorphism      234
Extension of scalars      174
Exterior power      507
Factor set      630
Factor set, trivial      630
Fiber product      437
Fiber square      434
Field extension, abelian      598
Field extension, cyclic      599
Field extension, maximal abelian      598
Field of fractions      209
Field, formally real      527
Field, global      598
Field, local      598
Field, ordered      527
Field, p-adic      580
Field, real closed      533
Filtration      89
Finite support      18
Fitting's lemma      77
Form, alternating      509
Form, bilinear      64
Form, multilinear      499
Form, nonsingular      67
Form, Pfister      538
Form, quadratic      69
Form, symmetric      64
Fractional ideal      226
Fractional ideal, principal      227
Free abelian monoid      74
Free algebra      490
Free group      401
Free module      17 21
Free monoid      24 490
Free ring      24
Full subcategory      5
functor      7
Functor, additive      161
Functor, contravariant      7
Functor, covariant      7
Functor, exact      160
Functor, forgetful      8
Functor, homology      11
Functor, homotopy      11
Functor, identity      8
Functor, inclusion      8
Functor, inverse      9
Functor, left exact      170
Functor, right exact      170
Fundamental theorem      485
G-map      136
G-set      13 136
G-set, transitive      137
Galois cohomology      634
General linear group      324
Graded components      494
Green ring      263
Grothendieck group of a category      84
Grothendieck group of a ring      86
Grothendieck group with respect to $\ast$      75
Grothendieck ring      148—153
Group ring      95
Group ring, twisted      630
Hensel's lemma      601
Homogeneous element      494 496
Homogeneous ideal      494
Homogeneous submodule      496
Horseshoe lemma      100
Hyperbolic plane      154
Ideal class group      226
Idempotent      43 268
Idempotent, central      162 276
Idempotent, centrally primitive      288
Idempotent, primitive      268
Index of an Azumaya algebra      624
Inert      240
Initial object      6
Inseparable degree      548
Integral closure      208
Integral element      207
Integral ideal      226
Integral ring extension      207
Integrally closed      209
Invariant basis number (IBN)      38
Inverse limit      582 593
Invertible ideal      212
Irreducible element      206
Isometric embedding      574
Isometric isomorphism      575
Isometry      64
Isomorphic      9 58
Isomorphism      6
Isomorphism class      58
Isomorphism class, restricted      58
Isomorphism of categories      9
Isomorphism, diagonal      440
Isotropic space      154
Isotropic vector      154
Jacobi identity      409
Jacobson density theorem      629
Jacobson radical      178
Just infinite      370
Kronecker dimension      533
Kronecker product      150
Krull dimension      128
Krull — Schmidt decomposition      76
Kummer's theorem      238
Lattice, RG-      259
Laurent polynomial      333
length function      70
Level of a field      529
Level of a subgroup      361
Lies above      230
linear combination      17
Linear map      1 2 145
Linear relation      17
Linearly dependent      17
Linearly independent      17
Linked idempotents      276
Little tensor      143 497
localization      186—187
Localization map      188
Localization sequence      481 483 608
Maschke's Theorem      274
Matrix completion      118
Matrix units      36 319
Matrix, change of basis      38 255
Matrix, diagonal      67 321 414
Matrix, elementary      320 324
Matrix, monomial      321 415
Matrix, partitioned      34
Matrix, permutation      320 414
Matrix, scalar      28
Matsumoto's Theorem      557 564
Maximal spectrum dimension      129
Maximum condition      215 648
Mayer — Vietoris sequence      469 517—518
Merkurjev — Suslin theorem      642
Metafunction      7
Metric      224 572
Milnor K-group      519
Milnor ring      519
Milnor's conjecture      531
Module, Artinian      649
Module, bilinear      64
Module, completely reducible      277
Module, cyclic      22
Module, faithful      207 628
Module, finitely generated (f.g.)      22
Module, flat      172
Module, Galois      264
Module, graded      496
Module, indecomposable      76
Module, injective      299
Module, left      1
Module, Noetherian      649
Module, nonsingular bilinear      67
Module, presentation of      22
Module, principal indecomposable      95 270
Module, projective      51
1 2
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