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Eisenbud D. — Commutative algebra with a view toward algebraic geometry
Eisenbud D. — Commutative algebra with a view toward algebraic geometry



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Название: Commutative algebra with a view toward algebraic geometry

Автор: 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: en

Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1995

Количество страниц: 797

Добавлена в каталог: 12.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$(I : J^{\infty})$, saturation      101 360 372
$Ass_RM$, Ass M, associated primes      89
$A^n$, $A_k^n$, $A^n(k)$, affine n-space      32
$B_IR$, blowup algebra      148
$B_JM$, blowup      149
$Der_R(S,M)$, R-linear derivations      383
$D^+(\mathfrak{M})$, derived category      680
$Fitt_j(M)$, Fitting ideal of M      493
$F^#$, map on functions      35
$f_{bottom}$, sum of low degree terms      356
$f_{\infty}$, sum of top degree terms      359
$GL_n(k)$, general linear group      24 38
$gr_1M$, $gr_jM$, associated graded ring or module      146
$Hom_R(M, N)$, homomorphisms      62
$H^0_I(M)$, annihilator of big power of I=zeroth local cohomology      100 683
$H_I^i(M)$, local cohomology      684
$H_M(s)$, Hilbert function of M      42 148
$id_AM$, injective dimension      529
$I_j\varphi$, determinantal ideal of $\varphi$      492
$I{-1}$ , inverse of a fractional ideal      253
$k * R^q[B]$, compositum of subrings      202
$K(R)^* = K{R)^{\times}$, group of units      248
$K(x_1,..., x_n)$ Koszul complex      428
$K(\mathfrak{M})$, $K^+(\mathfrak{M})$, homotopy categories of complexes      678
$k[x_1,... ,x_r]$, polynomial ring      13
$k[[x_1,..., x_n]]$, power series ring      179
$k[\Gamma]$, semigroup ring      139
$M[U^{-1}]$,$U^{-1}M$, localization      59
$m\bigotimes n$, image of (m, n) in $M\bigotimes_R N$      64
$M\bigotimes_R N$, tensor product      63 567
$m_{ij}$, $m_i/(GCD(m_i,m_j))$      322
$n_1\wedge\ldots\wedge n_t$, exterior product      349
$pd_RM$, projective dimension      470
$P^r$,$P^r_k$,$P^r(k)$, projective space      39
$R_0$, $R_1$, $S_0$, $S_1$, Serre's conditions      252
$R_P$, localization at a prime      60
$S(M)^*$ and S(M) as modules over one another      582
$SL_n(k)$, special linear group      24 38
$S^G$, ring of invariants      25 30
$S_d$, forms of degree d in $S = k[x_1,...,x_r]$      349
$S_R(M)$, (skew-)symmetric algebra of a graded module      569
$S_R(M)$, symmetric algebra      569
$Tor^R_i (M, N)$, derived functors of $\bigotimes$      159
$T_R(M)$, tensor algebra      569
$x^{(d)}$ divided power      579
$\bar{\har{\mathfrak{M}}}$, distinguished ideal of completion      181
$\bar{\mathfrak{F}}run(\mathfrak{B},\mathfrak{A})$, category of functors      698
$\Delta$,$\Delta^d$, diagonal (comultiplication)      576 578
$\hat{R}$,$\hat{R}_{\mathfrak{M}}$, completion      179
$\hat{Z}_p$, p-adic integers      182
$\I_{\infty}$, ideal at infinity      359
$\Lambda^i$, exterior power      349
$\mathscr{B}$, upper-triangular matrices (Borel subgroup)      349
$\mathscr{B}'$, lower-triangular matrices      349
$\mathscr{R}(R,I)$, Rees algebra      170
$\mathscr{U}$, identity plus strictly upper-triangular matrices (unipotent subgroup)      349
$\Omega^q_X$, exterior power of cotangent bundle      436
$\Omega_{S/R}$, Kaeher differentials      384
$\prec_p$, Gauss order on integers      352
$\sigma_{ij}$, monomial syzygy      320
$\wedge N$, or $\wedge_RN$, exterior algebra      423 569
$^rE$,$^rB$,$^rZ$,rth term, boundaries, cycles, of a spectral sequence      656
$^rE^{p,q} \Rightarrow_p H^{p+q}$ convergence of a spectral sequence      667
$^r_{hor}E$,$^r_{vert}E$, spectral sequences of a double complex      666
$^{\infty}E$, $^{\infty}B$, $^{\infty}Z$, limit term, boundaries, cycles, of a spectral sequence      656
$_{hor}G$, $_{vert}G$, parts of a double complex      661
(I: J), ideal quotient      15 360 372
1-generic matrices      604 605
5-lemma      635
5-term exact sequence      670
9-lemma      635
A(X), affine coordinate ring      33
Abel, N.H.      24
Abelian categories      614 690
Abhyankar, S.S.      306
Action of the symmetric group      25 575
Acyclicity lemma of Peskine — Szpiro      498
Additive functor      630
Adjoint functors      64 691—694
Affine algebraic set      32 76
Affine domain, integral closure of      263 292ff
Affine k-algebra      35 129
Affine n-space      32
Affine products      299
Affine rings      35 221 281ff
Affine schemes      36
Affine space is contractible      481
Akin, K.      585 591
Akizuki, Y.      261
Algebra      13
Algebraic curve      23
Algebraic extension of domains      130
Algebraic geometry, further readings      709
Algebraic independence      555 556
Algebraic integers      119
Algebraic subset      32
Algebraic variety      32
Algebraic vector bundle      616
Algorithm for monomial primary decomposition      111
Algorithms      317ff
Alternating matrix      588
Analogy with topology      566
Analysis situs      213
Analytic spread      276
Analytic view of normalization      128
Andre — Quillen homology      386
Andre, M.      386
Andreotti, A., and Salmon, P.      515
ann M, annihilator      15
Annihilation of Koszul cohomology      434
Annihilator      235 318 362 374
Annihilators and Fitting invariants      512 513
Antipode map of a Hopf algebra      576
Apery, R.      542
Apparently silly definition      506
Approximate root      183 209
Arabic, article in      576
Artin — Rees lemma      145 196 273 624
Artin — Tate proof of the Nullstellensatz      143
Artin, E.      143
Artinian      71 73 75 76 227 325
Ascending chain condition      27
Ascending chain condition on principal ideals      14
Assmus, E.F.      479 580
ASSOCIATED      89
Associated graded ring or module      146 319 342 356 372
Associated primes      87 89 108
Associated primes and free resolution      501
Associated primes and projective dimension      477
Associated primes of principal ideals      249 250
Associativity      11
Aufblasen (blow up)      149
Augmentation      576
Auslander — Buchsbaum      469 477 500
Auslander — Buchsbaum formula      469 475 477 485 499 590
Auslander's lemma      336
Auslander's transpose functor      641
Auslander, M.      643
Automorphisms      80
Average over a group      30
Avramov, L.      191 440
Avramov, L. and Halperin, S.      679
Axiom (computer algebra system)      375
Baer, R.      619 620
Balanced Tor      667
Base and fiber      236
Base change      391 494 570
Base point free pencil trick      442 505
Bases      556
Bases for TM, $\delta$M, SM, $\Lambda$M      572
Basis Theorem, Hilbert's      26
Basis, vector space, for a module      325
Bass' characterization of Noetherian rings      623
Bass' conjecture      485
Bass, H.      525
Bayer, D.      334 351 354 509
Bayer, D. and Stillman, M.      340
Beginner's binomial theorem      580
Bergman, G.      338
Berlekamp, E.      319
Bertini, E.      243
Betti numbers      639
Bezout, E.      305
Bialgebra      576 589
Big cell      369
Bigebra      576
Bilinear      63 567
Binary form      25
Binomial coefficients      54 352
Blowup algebra      148 243 319 372
Blowup of a regular sequence      441
Borel subgroup      349
Borel-fixed ideals      351 353 354
Bott's vanishing theorem      590
Boundary operators      611
Bounded rational functions are integral      128
Bouquet of circles      414
Bourbaki, N.      576
Branch of a plane curve      129 185
Brieskorn, E.      515
Brouwer, L.E.J.      214
Bruhat decomposition      369
Bruns, W.      377
Buchberger's algorithm      317 333
Buchberger's Criterion      332
Buchberger, B.      368
Buchsbaum ring      687
Buchsbaum — Rim complex      567 590
Buchsbaum — Rim multiplicity      591
Buchsbaum, D.A.      376 538 585 590 687
Buchsbaum, Eisenbud, Horrocks conjecture      502
Burch, L.      502
C(R), group of invertible ideals      256
C, complex numbers      12
Canonical bundle      519
Canonical curves, further readings      709
Canonical curves, trigonal      609
Canonical module      519 523 528
Canonical module as ideal      551
Canonical module, localization and completion      536
Canonical module, uniqueness and existence      534
Canonical ring of a variety      26
Cantor's one-to-one correspondence      214
Caratheodory's theorem      139
Cartan, H.      566
Cartier divisors      256 259 276 447
Castelnuovo — Mumford regularity      505 516
Castelnuovo — Mumford regularity, history of      510
Castelnuovo's base point free pencil trick      516
Castelnuovo, G.      442 505
Catalecticant matrix      604 605
Category      36 221 286 689ff
Catenary      453
Catenary, universally      286 288 312 453
Cauchy sequence      204
Cauchy, A.      307
Cayley numbers      482
Cayley — Bacharach      520 552 553
Cayley — Hamilton theorem      117 119 123
Cayley's $\Omega$-process      30
Cayley, A.      119 305 417 611 614
Center of a noncommutative ring      187
Chain complex      612
Chain of submodules      72
Change-of-rings spectral sequences      669 675 676
Characteristic of a field      205
Characteristic polynomial      371 575
Chasles, M.      553
Chern classes      44 488
Chern polynomial      488
Chevalley's Theorem      311
Chevalley's theorem, strong form      315
Chevalley, C.      48 57
Chinese remainder theorem      72 80
Choi, S.      515
Chow group      260 266
Chow(R), Chow group      260
Chow, W.L.      260
Circle, as a complex curve      52
Circular points at infinity      52 215
Class group      258
Classical groups      463
Classical topology      32 55 180 454
Clean primary decomposition      93
Clebsch, A.      24 215
Closed, in the Zariski topology      32
Coalgebra structure      576
CoCoA (computer algebra system)      375
codim, codimension      225
Codimension      225
Codimension and depth      448
Codimension one      247
Coefficient fields      189 190 201 205 397
Coequalizer      394 699 700
Cohen factorization      191
Cohen structure theorem      189 201
Cohen — Macaulay factor rings of codimension      2 544
Cohen — Macaulay implies universally catenary      453
Cohen — Macaulay module, existence of      465
Cohen — Macaulay property is local      452
Cohen — Macaulay ring      420 443 447—468 477 528 536 590 710
Cohen — Macaulay ring is equidimensional      454
Cohen — Macaulay rings of codimension 2      503
Cohen — Macaulay type      550
Cohen — Macaulay, extrinsic characterization      479
Cohen — Macaulayness in the geometric case      468
Cohen, I.S.      85
Coherent sheaf      44
Cohn, P.M.      338
Cohomology of coherent sheaves      44 467
Cokernel as a colimit      700
Colimits      391 394 697—699
Colimits of diagrams of commutative algebras      704
Colimits of diagrams of free modules      702
Colimits of diagrams of modules      700
Combinatorial invariant, generic initial ideal as      348
Commands in Macaulay      375
Commutative property      11
Compactification (completion) of affine space      40
Compactness      129
Comparison of local and global cohomology      673
Compatible orders      327 341
Complete intersection      462 537
Complete intersections are Cohen — Macaulay      455
Complete local ring      182 187
Complete set of orthogonal idempotents      13
Complete set of solutions      613
Complete with respect to an ideal      182
Completion of a regular local ring      484
Completion of a ring or module      179ff
Complex line or complex plane      215
Complex of differential forms      612
Complex of modules      45 417 611
Complex points      215
Complex, exactness of a      496
Complex, trivial      490
Complexes constructed by multilinear algebra      589
Complexes, constructions with      626ff
Complexity of computing Grobner bases      333
Composition series      72
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