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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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Compositum of subrings or subfields 557 Computation of differentials 387 Computation of Tor 160 Computation, further readings on 710 Computer algebra projects 375ff Comultiplication 576 Conductor 268 549 Conductor square 268 Cone 287 297 Conies 51 Conjecture of Bass 485 Conjecture of Buchsbaum, Eisenbud, Horrocks 502 Connected components 85 Connected in codimension 1 454 Connecting homomorphism on homology 428 631 Conormal module 177 387 Conormal sequence 387 Conormal sequence of a complete intersection 440 Constant rank, projective modules of 493 Constructible algebraic set 309 311 315 Constructing prime ideals 85 Constructive module theory 318 Contragredient representation 521 Convergence of a sequence in the Krull topology 192 Convergence of a spectral sequence 663 Convergent power series 184 Converse of the Principal Ideal Theorem 233 Convex cone 139 Coordinate geometry 23 Coordinate ring 33 Coprimary module 94 Coproducts 699 700 Cotangent bundle 383 388 Cotangent bundle functor 386 Cotangent bundle of projective space 436 Counit of a coalgebra 576 Counting constants 232 Covariant functor 160 Criteria for flatness 161ff Critical pairs in computation of Grobner basis 338 Cubic surface 504 Curve, Picard group of a 258 Cusp 51 de Rham complex 414 612 Dedekind domain 258 484 Dedekind, R. 22—24 87 256 Deformation, trivial 175 Deformations 175 176 410 Degree 44 300 Depth and codimension 448 Depth and dimension 686 Depth and flatness 460 Depth and localization 448 Depth and projection dimension 475 Depth and the vanishing of Ext 449 Depth is geometric 425 Depth of an ideal on a module 420 425 447 depth(I, M), length of maximal M-sequence 425 Depths of modules in an exact sequence 451 Derivations 383 583 Derived category 612 677 679 Derived exact couple 659 Derived functors 614 636 Desargues, G. 39 Descartes, R. 23 39 215 Descending chain condition 71 Descending chains of prime ideals 222 233 Descending multiplicative filtration 145 Determinant in multilinear algebra 565 Determinantal ideals, 106 112 244 371 Determinantal rings 463 710 Determinantal varieties 590 606 Determinate division 368 Determinate division algorithm 331 Devissage 308 Diagonal 300 576 Diagonal map 589 Diagram 697 Diagram of free modules over M 702 Diagrams and syzygies 634 Diameter and volume of a neighborhood 224 Differential basis 190 397 Differential module (=module with differential) 626 Differential modules, map of 627 Differential operator and inverse system 547 Differential operators 583 Differentials and colimits 394 Differentials and direct products 395 Differentials and localization 394 Differentials complete case 413 Differentials, Kaehler 383 611 dim, dimension 225 Dimension is a local property 218 Dimension of a graded ring 287 Dimension of a ring or module 43 213 225 246 555 Dimension of fibers 308 Dimension zero 227 546 Direct product 12 15 16 81 188 251 707 Direct product of Cohen — Macaulay rings 465 Direct sum, summand 15 31 47 571 700 707 Dirichlet, P.G.L. 22 Disconnected spectrum 85 Discrete valuation 248 Discrete valuation ring (DVR) 220 247 248 257 265 295 Disjoint union as colimit 707 Distinguished open set in Zariski topology 55 Distributive law 11 Divided Koszul relations 323 366 Divided power algebra 54 566 579 Divided powers and Pfaffians 588 Divided powers and the rational normal curve 587 Divisibility of binomials 352 Divisibility of monomials 320 Division algorithm 330 331 Division with remainder 318 330 Divisor 253 259 590 Divisors on a scroll 608 Divisors on rational normal curves 607 Dlementary symmetric function 25 296 Dlementary upper triangular matrices 349 Dolgachev, I. 503 Domain 12 150 166 Dominant morphism of varieties 288 Double complex 652 Double complex, third-quadrant 666 Double complexes with two rows 675 Dual socle generator 527 Duality 520 546 Duality for maximal Cohen — Macaulay modules 538 Dualizing functor 521 523 524 DVR 525; see discrete valuation ring Eagon — Northcott complex 567 590 606 Eagon, J. 590 Eakin, P. 625 Eckmann and Schopf 621 Eclater 149 Effective methods in commutative algebra 318 Effectivity in the Nullstellensatz 34 Eilenberg — MacLane spaces 566 Eilenberg, S. 690 Eisenstein's criterion 466 Elimination of elimination theory 306 Elimination order 357 371 Elimination theory 303 304 314 318 356 357 365 372 Elimination theory, main theorem of 303 314 ellipse 52 Elliptic curves, further readings 710 Elliptic normal curve 608 Elliptic quartic 456 Embedded deformation 175 Embedded prime or primary component 90 End(I) = Hom(I,I), endomorphism ring 265 Endomorphism ring 521 524 Enumerative geometry 232 Epimorphism 15 Equation of integral dependence 118 Equational criterion for flatness 164 Equations of an image 358 Equicharacteristic 189 205 Essential extension 622 Essential submodule 523 622 Etale topology, further readings 710 Euclidean 214 Euler characteristic 44 501 Euler, L. 22 305 Evans, E.G. 377 Exact couple 657 658 Exact functor 520 Exact sequence 16 611 626 Exact sequence of terms of low degree 670 Exact sequences of complexes 631 Exact triangle 658 Exactness of a complex 496 Exactness of limits 196 Excellent rings 192 293 295 Exceptional fiber 276 Exceptional set 149 Exchange property 556 Existence of valuation rings 264 Ext (extension functor) 363 642 611ff 644 Ext (extension functor) as an algebra 647 Exterior algebra 432 565 569 Exterior power 136 Exterior power of an ideal 574 Factoriality 14 98 125 483 514 552 Factoriality, homological characterization 514 Factoriality, hypersurface rings 515 Factoriality, regular local rings 480 Factoring polynomials; Hensel's lemma and 184 206—208 Factorization of polynomials over Z and Q 138 Families of graded modules 175 Family of projective plane curves 172 Family of varieties or algebras 155 Fermat's last theorem 22 Fermat, P. 23 39 Fiber of a blowup 276 Fiber square 267 Fibers 155 227 286 303 308 Fiberwise characterization of projectives 513 Field 11 Field theory 555ff Filtered colimit of free modules 703 Filtered colimits 701 Filtered colimits, exactness of 702 Filtered differential module 661 Filtered limits are not exact 708 Filtration, I-adic 145ff Filtration, I-stable 146 Filtration, m-adic 181 192 664 Filtration, stable 661 664 Filtrations 145 146 Finite free resolutions 54 474 Finite global dimension 477 Finite group 38 467 Finite injective dimension 530 531 Finite map 227 Finite over R 122 Finite projective dimension 475 477 Finite resolution 470 Finitely generated 17 135 Finitely presented 17 Finiteness of the integral closure 292 First-quadrant double complex 666 Fitting invariant and annihilator 511 Fitting invariant or ideal 404 489 492 494 496 Fitting's lemma 493 590 Five lemma 524 635 Five points in 370 Five-term exact sequence 670 Flags of linear subspaces 350 Flat modules as colimits of free modules 702 Flat modules, finitely presented 171 Flatness 66 68 137 155 163 165 183 237 303 342 703 Flatness and depth 460 Flatness and Hilbert functions 514 Flatness and regular sequences 468 Flatness of graded modules 174 Flattening stratifications 364 Flenner, H. 278 Forgetful functor 692 Form (= homogeneous polynomial) 13 Formal Nullstellensatz 49 Formal power series 179 Fourteenth problem, Hilbert's 26 Foxby, H.-B. 191 Fractional ideals 253 Fractions 59 Free algebra 692 Free and projective resolutions 617 Free module 16 615 Free presentation 17 492 Free product 707 Free resolution and associated primes 501 Free resolution of monomial ideals 439 Free resolutions 44 45 470 489 614 Free resolutions, linear 517 Function theory 23 Function, 150 153 functor 62 689 Functoriality 570 Fundamental group, further readings 710 Fundamental problem of invariant theory 25 Gaeta, F. 542 Gaffney, T. 591 Galligo, A. 351 Galois theory 290 293 Gauss' fundamental theorem of algebra 31 Gauss' lemma 109 125 126 Gauss, C.F. 21—23 31 352 Gaussian integers 21 GCD, greatest common divisor 320 General change of coordinates 338 Generalized local rings 510 Generalized principal ideal theorem 244 Generators and relations 17 Generic codimension 466 Generic fiber 286 Generic flatness 307 Generic freeness lemma, Grothendieck's 303 307 308 315 Generic initial ideal 348 351 Generic initial ideal, existence 349 Generic matrix 300 371 Generic smoothness 404 406 Geometric invariant theory 37 Geometric nature of depth 426 Germ of a variety 67 151 153 Germain, S. 23 Ghost candy 656 Gianni, P. 363 Gilmer, R. 218 Gimigliano, A. 503 Gin(I), generic initial ideal of f 349 gl dim R, global dimension 472 Global dimension 470 474 Gluing 84 Going down 237 238 243 289 Going up 129 227 Gordan, P. 26 337 367 Gorenstein rings 525 528 536 537 541 542 545 553 Gorenstein, D. 525
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