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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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Module, rank n projective 511 Module, reflexive 167 Module, right 2 Module, semisimple 277 Module, simple 41 178 232 Module, strongly indecomposable 77 Module, torsion 93 193 232 Monoid 5 23 72 Monoid homomorphism 23 72 Monoid ring 23 490 Monoid, cancellative 74 Nakayama's lemma 181 Natural isomorphism 11 Natural transformation 10 Negative K-group 485 Nilpotent element 179 Nilpotent ideal 180 Nilradical 180 Norm on Milnor K-theory 552 Norm residue homomorphism 643 Norm, algebra 332 547 Norm, field 236 548 Norm, functoriality of 548 555 Norm, ideal 235 241 Norm, isomorphism invariance of 547 555 Norm, transitivity of 548 Normal integral basis 264 Normalizes a subgroup 358 Null sequence 575 Number field 210 Number field, totally imaginary 393 Object 2 Odd maxima] ideal 597 Opposite ring 9 Orbit 137 Orthogonal group 81 Orthogonal idempotents 268 Orthogonal sum 64 Orthogonality relations 308—310 p-Adic integer 581 p-adic number 580 p-adic valuation 535 580 Perfect group 412 Permutation group 135 Picard group 514 Place 579 598 POSET 5 Prestability 132 Prime ideal 128 214 Prime spot 598 Primitive solution 585 Product formula 591 595 598 605—606 Product of rings 280 Projection 54 Projection formula 559 Projective basis 54 Projective class group 106 Pullback 438 Quadratic reciprocity 395 591 Quaternions 264 631 Quillen K-theory 446 521 524 608 642 Radical ideal 180 Ramification index 231 Ramify 240 Rank, absolute stable 129 Rank, extended free 176 Rank, free 38 Rank, generalized 87 Rank, local 200 511 516 Rank, presentation 97 Rank, stable 121 122 343 Rank, stably free 108 Rank, torsion free 103 Reciprocity law 580 591 Reciprocity map 599 Reciprocity, q- 561 Reflection 81 Relative exact sequence 431 449—450 Relative K-group 361 445 447 Relative, Whitehead lemma 358 Relators 22 25 401 Representation of a form 65 Representation of a group 253 Representation of a linear map 30 Representation of a ring 264 Representation ring 263 Representation, absolutely irreducible 291 Representation, afforded by a module 257 Representation, defined over a field 294 Representation, faithful 253 Representation, full 290 Representation, indecomposable 262 Representation, irreducible 262 Representation, linear 253 Representation, matrix 30 65 253 Representation, permutation 135 Representation, projective 262 Representation, regular 257 Representation, simple 262 Representation, trivial 257 Residue, classes 231 Residue, degree 231 Residue, field 231 534 Residue, ring 231 Resolution 99 Resolution Theorem 102 Resolution, finite 99 Resolution, projective 99 Restriction of scalars 163 Rim square 436 Rim's Theorem 472 Ring 1 Ring of arithmetic type 393 Ring presentation 25 Ring, Artinian 287 649 Ring, associated graded 496 532 Ring, discrete valuation 221 534 Ring, division 41 Ring, factorial 206 Ring, generalized euclidean 322 Ring, graded 494 Ring, hereditary 248 Ring, J-noetherian 129 Ring, left regular 99 Ring, local 183 Ring, Noetherian 129 649 Ring, semilocal 201 220 Ring, semisimple 278 Ring, simple 287 Ring, truncated polynomial 496 Row equivalent, t- 321 Sandwich condition 367 Schanuel's lemma 97 Schur multiplier 412 Schur's lemma 281 Self-centralizing subfield 642 Semigroup 71 Semigroup homomorphism 71 Semigroup, cancellative 74 Semiring 133 Semiring homomorphism 134 Separable degree 548 Shorten, absolute n- 129 Shorten, n- 121 122 Shortened, can be 121 128 Similarity class of Azumaya algebras 620 Similarity class of matrices 60—63 Similarity class of matrix representations 255 Similarity class, stable 60 Simple component 288 Singular homology 11 Skipping lemma 122 Skolem — Noether theorem 629 Snake lemma 50 Snake lemma for groups 363 Solvable group 327 Spectrum, maximal 129 Spectrum, prime 191 Sphere 81 Split Azumaya algebra 624 626 Split prime 240 Split, maximally 626 Splitting field of a group 293 315 Splitting field of an Azumaya algebra 624 Square class 158 583 Stability for 104—132 Stability for 342—356 Stability for relative 364 Stability, injective 348—356 Stability, surjective 343—348 Stabilization map 343 Stabilizer 137 Stable normal structure theorem 360 Stable range 105 Stably equivalent 105 Stably free 108 Stably isomorphic 105 Standard relations 402 Stein relativization 432 Steinberg group 403 Steinberg group, relative 432 448—449 Steinitz theorem 243 Structure constants 493 Stufe 529 Subdirect product 468 Subdirect sum 369 Sum of submodules 267 Symbol map 564 Symbol, 2-adic 586—589 Symbol, Dennis — Stein 565 Symbol, extended tame 542 Symbol, Hilbert 570 583 586 Symbol, Legendre 584 Symbol, Mennicke 377 Symbol, power norm residue 599 Symbol, power residue 393—394 Symbol, real 395 526 Symbol, Steinberg 420 Symbol, tame 536 Symmetric power 505 Tame kernel 608 Tensor power 500 Tensor product of algebras 611 Tensor product of bimodules 144—146 Tensor product of matrices 150 Tensor product of modules 142 497—498 Tensor product, internal 614 Terminal object 6 Torsion element 40 604 Torsion free 40 Torsion subgroup 604 Totally ordered abelian group 527 Trace 217 301 Transfer map 332 TRANSPOSE 37 Transpose, skew 390 Transvection, elementary 320 Transvection, elementary row 320 Triangle inequality 572 Twist 630 Underlying division algebra 620 Underlying permutation 415 Unimodular, left 115 Unimodular, right 115 UNIT 8 Unstable normal structure theorem 368 Valuation 533 Valuation, discrete 220—221 534 Valuation, p-adic 535 Value group 572 Valued field 572 Valued field, complete 574 Weakly finite 113 Weakly n-finite 113 Wedderburn — Artin theorem 283 Wedderburn's Theorem 428—429 636 Weil reciprocity 549 Weyl group 422 Whitehead group 329 Whitehead lemma 325 Wild kernel 606 Witt index 158 Witt ring 156 Witt — Grothendieck group 82 Witt — Grothendieck ring 153 Zero divisor 40
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