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Magurn B.A. — An algebraic introduction to k-theory
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Название: An algebraic introduction to k-theoryАвтор: 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: Рубрика: Математика /Серия: Сделано в холле Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2002Количество страниц: 676Добавлена в каталог: 17.05.2011Операции: Положить на полку |
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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 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 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
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