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Eisenbud D. — Commutative algebra with a view toward algebraic geometry
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Название: Commutative algebra with a view toward algebraic geometryАвтор: Eisenbud D. Аннотация: Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algebraic geometry. To help beginners, the essential ideals from algebraic geometry are treated from scratch. Appendices on homological algebra, multilinear algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text.
Язык: Рубрика: Математика /Алгебра /Алгебраическая геометрия /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1995Количество страниц: 797Добавлена в каталог: 12.03.2005Операции: Положить на полку |
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Maple (computer algebra system) 375 Mapping cone 427 428 650 Mapping cone and double complexes 650 Mapping cones revisited 657 Mapping cones to spectral sequences 650 Maps and homotopies of complexes 627 Maps from power series rings 198 Maps from projective to acyclic complexes 491 Massey 657 Mathematica (computer algebra system) 375 Matrices of linear forms 604 606 Matroid 556 Max Noether's AF+BG theorem 466 max-Spec R, maximal spectrum 54 maximal Cohen — Macaulay modules 529 599 Maximal ideals 12 141 Maximal M-sequence 430 Maximal minors 463 Maximal regular sequences 424 Maximal spectrum (max-spec) 54 Maximal subfields 205 Menger, K. 214 Minimal complex 472 Minimal free resolution 478 Minimal free resolution is unique 490 491 Minimal generators of monomial ideal 320 Minimal Groebner basis 329 367 Minimal injective resolution 529 623 Minimal map 472 Minimal primary decomposition 95 Minimal prime 77 Minimal primes of I 47 Minimal resolutions 469 Minimal set of generators 472 Minor, 244 Minor, principal 369 Minors of a matrix 107 244 300 301 370 465 565 Minors, 53 Miyata, T. 649 Model theory 142 Module of homomorphisms 318 Modules 15 Modules over a Dedekind domain 484 Moduli 38 Moebius, A.F. 39 Moller, H.M. 333 Monic polynomial 117 126 138 Monomial 13 139 319 349 Monomial basis 323 Monomial curve singularities 379 Monomial ideal 111 319 365 Monomial orders 323 324 Monomial submodule 319 Monomorphism 15 Mora, T. 333 Mori, S. 26 263 Morin, B. 403 Morphism 35 304 689 697 Mourrain, B. 371 Multilinear algebra 565 Multiple point on a line 178 Multiplicatively closed set 59 70 Multiplicity 259 274 276 277 300 Multiplicity in primary decomposition 102 Mumford, D. 38 505 506 516 Nagata rings 294 Nagata's altitude formula 289 299 Nagata's factoriality lemma 483 Nagata, M. 26 105 184 263 483 Nakayama's lemma 124 129 135 136 203 241 298 299 Nakayama's lemma, homological version 473 Natural transformation 689 690 698 Neighborhoods 57 156 Newton's method 183 Newton, I. 23 295 305 Nielsen, H.A. 591 Nilpotent 33 36 71 Nilpotents do not affect dimension 219 Nine lemma 635 Nodal plane cubic 185 Noether normalization 221 281 283 298 308 460 Noether normalization with separating transcendence base 401 Noether — Lefschetz theorem 515 Noether, E. 23 27 47 49 57 87 118 127 217 231 256 296 612 Noetherian 27 28 46 62 85 183 192 229 265 294 625 Nonconstructive fashion 317 Nonhomogeneous Grobner bases 372 Nonsingular 249 Nontrivial idempotent 85 Nonunique resolution 515 Nonzerodi visor 12 163 Normal cone 441 Normal cone of a regular sequence 441 Normal domain 118 251 Normal expression 349 Normal module 177 Normal ring 125 126 138 249 251 289 Normal vector field 177 Normalization 118 126 128 137 141 251 364 North pole 50 Nullstellensatz 31 33 38 131 141 311 413 Number theory 21 Numbers of generators of powers 276 Objects of a category 689 Obstruction to generating M 494 One-generic matrices 604 605 Open morphism 238 299 Orientation 434 Orthogonal idempotents 13 187 Orthonormal basis 466 Osculating plane 380 p-adic integers 184 p-adic numbers 179 182 p-adic unit 209 p-bases 559 p-basis 190 Parallel lines meet at infinity 39 Parameter ideal 234 235 272 275 278 Pardue, K. 348 Partitions 585 Partitions of unity 83 Pascal, B. 553 Pathologies 218 221 229 Peano, G. 214 Perfect fields 190 Perfect ideals 485 Periodic polynomial 245 Permutation of a regular sequence 422 Perpendicular subspace 444 Persymmetric matrix 604 Peskine and Szpiro 498 539 Peskine, C. 334 540 591 Pfaff 588 Pfaffians 463 503 588 Pic(i2), group of isomorphism classes of invertible ideals 255 Picard group 255 258 Pit 260; see Principal ideal theorem Pluecker, J. 24 39 Poincare, H. 213 612 Points at infinity 40 Pole, order of a 246 Polynomial equations 34 Polynomial function 31 33 Polynomial map 35 Polynomial ring 394 Polynomial with periodic coefficients 245 280 Polynomial, factoring 305 Portnoy's Complaint 709 Positive characteristic, further readings 710 Positive cone 366 Power series 205 Power series ring 189 193 198 Pragacz, P. 591 Priddy's generalized Koszul complex 444 445 Priddy, S. 338 Primary component 95 Primary decomposition 23 87 94 363 477 710 Primary decomposition and projective dimension 477 Primary decomposition, (non)-uniqueness of 102 Primary decomposition, geometry of 103 Primary decomposition, uniqueness of 111 Primary ideal or submodule 94 111 Prime avoidance 90 113 Prime element 14 Prime ideal 12 54 60 70 111 243 Prime in a graded ring 297 Primes in an integral extension 129 Primes in polynomial rings 297 Primes, minimal over I 47 90 Primitive polynomial 109 Principal ideal domain 13 54 137 163 Principal ideal ring 228 Principal ideal theorem 231ff Principal ideal theorem for depth 449 Principal ideal theorem, converse to 233 Problems in computation 364 Products of algebraic sets 299 Products of domains 78 Products of rings, expression by idempotents 85 Projection from a product to a factor 299 304 318 Projection to a quotient 37 Projective algebraic set 40 Projective closure 41 359 Projective closure by saturation 372 Projective dimension 469 470 Projective dimension and associated primes 477 Projective dimension and depth 451 475 Projective dimension and primary decomposition 477 Projective geometry 51 Projective module 136 137 265 267 364 471 615 Projective modules, characterization of 616 Projective morphism 166 Projective of constant rank 493 496 513 Projective plane, complex 215 Projective products 299 Projective r-space 39 Projective resolution 469 626 Projective resolution functor 682 Projective varieties 39 304 464 Proper ideal 12 Proper morphism 118 129 166 300 305 315 Pseudo-reflection 48 Pseudogeometric 192 pth power map 201 Puiseux, V. 295 Pullback 362 Pure codimension 1 257 Pure transcendental extension 555 Q, rational numbers 12 Quadratic algebra 444 Quadratic relations 444 Quadric, rank of 467 Quadrics 51 459 466 Quasi-isomorphism of complexes 655 680 Quaternions 482 Quick and dirty proof of the Nullstellensatz 142 Quillen, D. 386 481 Quism 482; see quasi-isomorphism Quotient by a group 37 296 297 Quotient by an ideal (I:J), computing 15 374 Quotient on division 320 R, real numbers 12 R-bilinear 160 R-linear 691 R-sequence 241 R0 252 266 R1 252 Rabinowitch's trick 132 142 rad /, radical of an ideal 71 Radical 33 71 111 137 Radical complete intersection 415 Radical of an ideal, computing 363 Rank of a module 16 135 261 Rank of a quadric 467 Rank of an ideal, synomym for codimension 226 Rao, A.P. 540 Rational convex polyhedral cone 464 Rational curves, computer project 378 Rational function 251 555 Rational map 266 Rational normal curve 380 587 606 650 Rational quartic in 466 Ratliff, L. 289 Reduced Groebner basis 329 367 Reduced ring 33 251 266 562 Reduction to the diagonal 277 299—301 Reduction to the join 301 Reductive group 584 Rees algebra 167 342 Rees, D. 591 Refined by 326 Reflection of isomorphism 203 reg M, Castelnuovo — Mumford regularity 505 Regular function 33 251 Regular in codimension 1 249 266 Regular local ring 191 240 469 474 484 Regular local ring is factorial 483 Regular local ring, Jacobian criterion for 402 Regular local rings and regular sequences 465 Regular local rings are Cohen — Macaulay 462 Regular map 35 Regular parameter 247 Regular sequence 173 241 243 246 419 426 437 441 465 Regular sequence in any order, generation by 438 Regular sequences and flatness 468 Regular sequences permutability 422 438 Regular sequences, characterization of 517 Regular system of parameters 240 Regularity and hyperplane sections 508 Regularity of generic initial ideals 509 Regularity, Castelnuovo — Mumford 378 505 516 Relative cotangent bundle 389 Relative cotangent sequence 386 Relative tangent bundle 389 Remainder on division 330 Removable singularities theorem 251 Representable functors 695 Representation theory of GL(V), SL(V) 566 584 Representation theory, success of 479 480 Residue class field 60 Residue map 548 Resolution 617 626 Resolution of an ideal from a factor ring 654 Resolution of singularities 249 Resolutions and linkage 552 Resolutions of complexes 676 Restricted tensor product 392 705 Resultant, Sylvester's 307 Retraction 206 Reverse lexicographic deformation of 3 points 346 Reverse lexicographic deformation of conic 347 Reverse lexicographic order 326 328 339 344 Riemann sphere 215 Riemann surfaces and algebraic curves 215 Riemann — Roch theorem 44 456 Riemann, G.B. 24 215 Right adjoint 691 Right exact 65 Right exact functor 636 Right exact sequence 63
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