Verfügbarkeit: Commutative algebra with a view toward algebraic geometry

Commutative algebra with a view toward algebraic geometry:

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Bibliographische Detailangaben
1. Verfasser:	Eisenbud, David 1947- (VerfasserIn)
Format:	Buch
Sprache:	German
Veröffentlicht:	New York [u.a.] Springer 2004
Ausgabe:	[7. print]
Schriftenreihe:	Graduate texts in mathematics 150
Schlagworte:	Anéis e álgebras comutativos Geometria algébrica Commutative algebra Geometry, Algebraic Algebraische Geometrie Kommutative Algebra
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	Literaturverz. S. 757 - 774
Beschreibung:	XVI, 797 S. graph. Darst.
ISBN:	3540942696 0387942696 0387942688 9780387942698

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Datensatz im Suchindex

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adam_text	Contents Introduction 1 Advice for the Beginner ....................... 2 Information for the Expert ..................... 2 Prerequisites ............................. 6 Sources ................................ 6 Courses ................................ 7 A First Course ......................... 7 A Second Course ....................... 8 Acknowledgements .......................... 9 0 Elementary Definitions 11 0.1 Rings and Ideals ........................ 11 0.2 Unique Factorization ..................... 13 0.3 Modules ............................ 15 I Basic Constructions 19 1 Roots of Commutative Algebra 21 1.1 Number Theory ........................ 21 1.2 Algebraic Curves and Function Theory ........... 23 1.3 Invariant Theory ....................... 24 1.4 The Basis Theorem ...................... 27 1.4.1 Finite Generation of Invariants ........... 29 viii Contents 1.5 Graded Rings ......................... i0 1.6 Algebra and Geometry: The Nullstellensatz ........ 31 1.7 Geometric Invariant Theory ................. 37 1.8 Projective Varieties ...................... 39 1.9 Hubert Functions and Polynomials ............. 42 1.10 Free Resolutions and the Syzygy Theorem ......... 44 1.11 Exercises ............................ 46 Noetherian Rings and Modules ............. 4*> An Analysis of Hubert s Finiteness Argument .... 47 Some Rings of Invariants ................ 48 Algebra and Geometry ................. 4^ Graded Rings and Projective Geometry ........ 52 Hubert Functions .................... 53 Free Resolutions ..................... 54 Spec, max-Spec, and the Zariski Topology ...... 54 2 Localization 57 2.1 Fractions ............................ 59 2.2 Hom and Tensor ........................ 62 2.3 The Construction of Primes ................. 70 2.4 Rings and Modules of Finite Length ............. 71 2.5 Products of Domains ..................... 78 2.6 Exercises ............................ 78 Z-graded Rings and Their Localizations ........ 81 Partitions of Unity .................... 83 Gluing .......................... 83 Constructing Primes ................... 84 Idempotents, Products, and Connected Components . 85 3 Associated Primes and Primary Decomposition 87 3.1 Associated Primes ....................... 89 3.2 Prime Avoidance ....................... 90 3.3 Primary Decomposition .................... 94 3.4 Primary Decomposition and Factoriality .......... 98 3.5 Primary Decomposition in the Graded Case ........ 99 3.6 Extracting Information from Primary Decomposition . . . 100 3.7 Why Primary Decomposition Is Not Unique ........ 102 3.8 Geometric Interpretation of Primary Decomposition .... 103 3.9 Symbolic Powers and Functions Vanishing to High Order . 105 3.9.1 A Determinantal Example .............. 107 3.10 Exercises ............................ 109 General Graded Primary Decomposition ....... 110 Primary Decomposition of Monomial Ideals ...... Ill The Question of Uniqueness .............. 112 Determinantal Ideals .................. 113 Contents ix Total Quotients..................... 113 Prime Avoidance ..................... 114 4 Integral Dependence and the Nullstellensatz 117 4.1 The Cayley-Hamilton Theorem and Nakayama s Lemma . 119 4.2 Normal Domains and the Normalization Process ...... 125 4.3 Normalization in the Analytic Case ............. 128 4.4 Primes in an Integral Extension ............... 129 4.5 The Nullstellensatz...................... 131 4.6 Exercises ............................ 135 Nakayama s Lemma ................... 136 Projective Modules and Locally Free Modules .... 136 Integral Closure of Ideals ................ 137 Normalization ...................... 138 Normalization and Convexity .............. 139 Nullstellensatz...................... 142 Three More Proofs of the Nullstellensatz....... 142 5 Filtrations and the Artin-Rees Lemma 147 5.1 Associated Graded Rings and Modules ........... 148 5.2 The Blowup Algebra ..................... 150 5.3 The Krull Intersection Theorem ............... 152 5.4 The Tangent Cone ...................... 153 5.5 Exercises ............................ 154 6 Flat Families 157 6.1 Elementary Examples ..................... 159 6.2 Introduction to Tor ...................... 161 6.3 Criteria for Flatness ...................... 162 6.4 The Local Criterion for Flatness ............... 167 6.5 The Rees Algebra ....................... 171 6.6 Exercises ............................ 172 Flat Families of Graded Modules ............ 175 Embedded First-Order Deformations ......... 176 7 Completions and Hensel s Lemma 181 7.1 Examples and Definitions ................... 181 7.2 The Utility of Completions .................. 184 7.3 Lifting Idempotents ...................... 188 7.4 Cohen Structure Theory and Coefficient Fields ....... 191 7.5 Basic Properties of Completion ............... 194 7.6 Maps from Power Series Rings ................ 200 7.7 Exercises ............................ 205 Modules Whose Completions Are Isomorphic ..... 205 The Krull Topology and Cauchy Sequences ...... 206 Contents Completions from Power Series .............207 Coefficient Fields .................... 207 Other Versions of Henseľs Lemma ...........208 II Dimension Theory 213 8 Introduction to Dimension Theory 215 8.1 Axioms for Dimension .................... 220 8.2 Other Characterizations of Dimension ............ 222 8.2.1 Affine Rings and Noether Normalization ...... 223 8.2.2 Systems of Parameters and Krull s Principal Ideal Theorem ....................... 224 8.2.3 The Degree of the Hubert Polynomial ....... 225 9 Fundamental Definitions of Dimension Theory 227 9.1 Dimension Zero ........................229 9.2 Exercises ............................230 10 The Principal Ideal Theorem and Systems of Parameters 233 10.1 Systems of Parameters and Ideals of Finite Colength . . . 236 10.2 Dimension of Base and Fiber ................. 238 10.3 Regular Local Rings ...................... 242 10.4 Exercises ............................ 244 Determinantal Ideals ..................246 Hubert Series of a Graded Module ...........247 11 Dimension and Codimension One 251 11.1 Discrete Valuation Rings ................... 251 11.2 Normal Rings and Serre s Criterion ............. 253 11.3 Invertible Modules ...................... 257 11.4 Unique Factorization of Codimension-One Ideals ...... 260 11.5 Divisors and Multiplicities .................. 262 11.6 Multiplicity of Principal Ideals ................ 265 11.7 Exercises ............................ 268 Valuation Rings ..................... 268 The Grothendieck Ring ................. 269 12 Dimension and Hilbert-Samuel Polynomials 275 12.1 Hilbert-Samuel Functions ...................276 12.2 Exercises ............................279 Analytic Spread and the Fiber of a Blowup ......280 Multiplicities .......................280 Hubert Series ......................284 Contents xi 13 The Dimension of Affine Rings 285 13.1 Noether Normalization .................... 285 13.2 The Nullstellensatz...................... 296 13.3 Finiteness of the Integral Closure.............. 297 13.4 Exercises ............................ 300 Quotients by Finite Groups ............... 300 Primes in Polynomial Rings .............. 301 Dimension in the Graded Case ............. 302 Noether Normalization in the Complete Case ..... 303 Products and Reduction to the Diagonal ....... 304 Equational Characterization of Systems of Parameters 306 14 Elimination Theory, Generic Freeness, and the Dimension of Fibers 307 14.1 Elimination Theory ...................... 307 14.2 Generic Freeness ........................ 312 14.3 The Dimension of Fibers ................... 313 14.4 Exercises ............................ 318 Elimination Theory ................... 318 15 Gröbner Bases 321 Constructive Module Theory .............. 322 Elimination Theory ................... 322 15.1 Monomials and Terms .................... 323 151.1 Hubert Function and Polynomial .......... 324 15.1.2 Syzygies of Monomial Submodules ......... 326 15.2 Monomial Orders ....................... 327 15.3 The Division Algorithm .................... 333 15.4 Gröbner Bases ......................... 335 15.5 Syzygies ............................ 337 15.6 History of Gröbner Bases ................... 340 15.7 A Property of Reverse Lexicographic Order ........ 342 15.8 Gröbner Bases and Flat Families .............. 345 15.9 Generic Initial Ideals ..................... 351 15.9.1 Existence of the Generic Initial Ideal ........ 353 15.9.2 The Generic Initial Ideal is Borel-Fixed ...... 354 15.9.3 The Nature of Borel-Fixed Ideals .......... 355 15.10 Applications .......................... 358 15.10.1 Ideal Membership .................. 359 15.10.2 Hubert Function and Polynomial .......... 359 15.10.3 Associated Graded Ring ............... 360 15.10.4 Elimination ...................... 361 15.10.5 Projective Closure and Ideal at Infinity ...... 362 15.10.6 Saturation ...................... 363 15.10.7 Lifting Homomorphisms ............... 364 xii Contents 15.10.8 Syzygies and Constructive Module Theory .... 365 15.10.9 What s Left? ..................... 367 15.11 Exercises ............................ 368 15.12 Appendix: Some Computer Algebra Projects ........ 378 Project 1. Zero-dimensional Gorenstein Ideals .... 376 Project 2. Factoring Out a General Element from an sth Syzygy .................. 377 Project 3. Resolutions over Hypersurfaces ....... 377 Project 4. Rational Curves of Degree Г+ 1 in Pr. .. 378 Project 5. Regularity of Rational Curves ....... 378 Project 6. Some Monomial Curve Singularities .... 379 Project 7. Some Interesting Prime Ideals ....... 379 16 Modules of Differentials 385 16.1 Computation of Differentials ................. 390 16.2 Differentials and the Cotangent Bundle ........... 390 16.3 Colimits and Localization .................. 393 16.4 Tangent Vector Fields and Infinitesimal Morphisms .... 398 16.5 Differentials and Field Extensions .............. 400 16.6 Jacobian Criterion for Regularity .............. 404 16.7 Smoothness and Generic Smoothness ............ 407 16.8 Appendix: Another Construction of Kahler Differentials . 410 16.9 Exercises ............................412 III Homological Methods 421 17 Regular Sequences and the Koszul Complex 423 17.1 Koszul Complexes of Lengths 1 and 2............ 424 17.2 Koszul Complexes in General ................ 427 17.3 Building the Koszul Complex from Parts .......... 431 17.4 Duality and Homotopies ................... 436 17.5 The Koszul Complex and the Cotangent Bundle of Projective Space ........................ 440 17.6 Exercises ............................ 441 Free Resolutions of Monomial Ideals .......... 443 Conormal Sequence of a Complete Intersection .... 444 Regular Sequences Are Like Sequences of Variables . 445 Blowup Algebra and Normal Cone of a Regular Sequence ................... 445 Geometric Contexts of the Koszul Complex ......447 Contents xiii 18 Depth, Codimension, and Cohen-Macaulay Rings 451 18.1 Depth ............................. 451 18.1.1 Depth and the Vanishing of Ext .......... 453 18.2 Cohen-Macaulay Rings .................... 455 18.3 Proving Primeness with Serre s Criterion .......... 461 18.4 Flatness and Depth ...................... 464 18.5 Some Examples ........................ 466 18.6 Exercises ............................ 469 19 Homological Theory of Regular Local Rings 473 19.1 Projective Dimension and Minimal Resolutions ...... 473 19.2 Global Dimension and the Syzygy Theorem ........ 478 19.3 Depth and Projective Dimension: The Auslander- Buchsbaum Formula ..................... 479 19.4 Stably Free Modules and Factoriality of Regular Local Rings .......................... 484 19.5 Exercises ............................ 488 Regular Rings ...................... 488 Modules over a Dedekind Domain ........... 488 The Auslander-Buchsbaum Formula .......... 489 Projective Dimension and Cohen-Macaulay Rings . . 489 Hubert Function and Grothendieck Group ...... 490 The Chern Polynomial ................. 492 20 Free Resolutions and Fitting Invariants 493 20.1 The Uniqueness of Free Resolutions ............. 494 20.2 Fitting Ideals ......................... 496 20.3 What Makes a Complex Exact? ............... 500 20.4 The Hilbert-Burch Theorem ................. 506 20.4.1 Cubic Surfaces and Sextuples of Points in the Plane ......................... 508 20.5 Castelnuovo-Mumford Regularity .............. 509 20.5.1 Regularity and Hyperplane Sections ........ 513 20.5.2 Regularity of Generic Initial Ideals ......... 514 20.5.3 Historical Notes on Regularity ........... 514 20.6 Exercises ............................ 515 Fitting Ideals and the Structure of Modules ..... 515 Projectives of Constant Rank .............. 518 Castelnuovo-Mumford Regularity ........... 521 21 Duality, Canonical Modules, and Gorenstein Rings 523 21.1 Duality for Modules of Finite Length ............ 524 21.2 Zero-Dimensional Gorenstein Rings ............. 529 21.3 Canonical Modules and Gorenstein Rings in Higher Dimension ........................... 532 xiv Contents 21.4 Maximal Cohen-Macaulay Modules............. 533 21.5 Modules of Finite Injective Dimension ........... 534 21.6 Uniqueness and (Often) Existence .............. 538 21.7 Localization and Completion of the Canonical Module . . 540 21.8 Complete Intersections and Other Gorenstein Rings .... 541 21.9 Duality for Maximal Cohen-Macaulay Modules ...... 542 21.10 Linkage ............................-543 21.11 Duality in the Graded Case ................. 549 21.12 Exercises ............................ 550 The Zero-Dimensional Case and Duality ....... 550 Higher Dimension .................... 552 The Canonical Module as Ideal ............. 555 Linkage and the Cayley-Bacharach Theorem ..... 556 Appendix 1 Field Theory 555 Al.l Transcendence Degree .....................561 A1.2 Separability ..........................563 A1.3 p-Bases .............................565 Al.3.1 Exercises .......................568 Appendix 2 Multilinear Algebra 565 A2.1 Introduction .......................... 571 A2.2 Tensor Products ........................ 573 A2.3 Symmetric and Exterior Algebras .............. 574 A2.3.1 Bases ......................... 578 A2.3.2 Exercises ....................... 580 A2.4 Coalgebra Structures and Divided Powers ......... 581 A2.4.1 S(M)· and S(M) as Modules over One Another . 582 A2.5 Schur Functors ......................... 590 A2.5.1 Exercises ....................... 594 A2.6 Complexes Constructed by Multilinear Algebra ...... 596 A2.6.1 Strands of the Koszul Complex ........... 597 A2.6.2 Exercises ....................... 609 Appendix 3 Homological Algebra 611 A3.1 Introduction .......................... 617 Part I: Resolutions and Derived Functors .......... 614 A3.2 Free and Projective Modules ................. 621 A3.3 Free and Projective Resolutions ............... 623 A3.4 Injective Modules and Resolutions .............. 624 A3.4.1 Exercises ....................... 630 Injective Envelopes ................... 630 Injective Modules over Noetherian Rings ....... 630 A3.5 Basic Constructions with Complexes ............ 632 A3.5.1 Notation and Definitions .............. 632 Contents xv A3.6 Maps and Homotopies of Complexes ............ 633 A3.7 Exact Sequences of Complexes ................ 637 АЗ^Л Exercises ....................... 638 A3.8 The Long Exact Sequence in Homology ........... 639 A3.8.1 Exercises ....................... 640 Diagrams and Syzygies ................. 640 A3.9 Derived Functors ....................... 643 A3.9.1 Exercise on Derived Functors ............ 645 АЗ.ЮТог ............................... 646 A3.10.1 Exercises: Tor .................... 646 A3.ll Ext ............................... 649 АЗ.П.І Exercises: Ext .................... 651 A3.11.2 Local Cohomology .................. 656 Part II: From Mapping Cones to Spectral Sequences . . 650 A3.12 The Mapping Cone and Double Complexes ......... 656 A3.12.1 Exercises: Mapping Cones and Double Complexes 660 A3.13 Spectral Sequences ...................... 663 A3.13.1 Mapping Cones Revisited .............. 664 A3.13.2 Exact Couples .................... 665 A3.13.3 Filtered Differential Modules and Complexes ... 668 A3.13.4 The Spectral Sequence of a Double Complex ... 671 A3.13.5 Exact Sequence of Terms of Low Degree ...... 677 A3.13.6 Exercises on Spectral Sequences .......... 678 A3.14 Derived Categories ...................... 684 A3.14.1 Step One: The Homotopy Category of Complexes 685 A3.14.2 Step Two: The Derived Category .......... 686 A3.14.3 Exercises on the Derived Category ......... 688 Appendix 4 Λ Sketch of Local Cohomology 683 A4.1 Local Cohomology and Global Cohomology ........ 693 A4.2 Local Duality ......................... 694 A4.3 Depth and Dimension ..................... 695 Appendix 5 Category Theory 689 A5.1 Categories, Functors, and Natural Transformations .... 697 A5.2 Adjoint Functors ....................... 699 A5.2.1 Uniqueness ...................... 700 A5.2.2 Some Examples ................... 700 A5.2.3 Another Characterization of Adjoints ....... 701 A5.2.4 Adjoints and Limits ................. 702 A5.3 Representable Functors and Yoneda s Lemma ....... 703 xvi Contents Appendix 6 Limits and Colimits 697 A6.1 Colimits in the Category of Modules............708 A6.2 Flat Modules as Colimits of Free Modules.........711 A6.3 Colimits in the Category of Commutative Algebras .... 713 A6.4 Exercises ............................715 Appendix 7 Where Next? 709 Hints and Solutions for Selected1 Exercises 711 References 757 Index of Notation 775 Index 779
adam_txt	Contents Introduction 1 Advice for the Beginner . 2 Information for the Expert . 2 Prerequisites . 6 Sources . 6 Courses . 7 A First Course . 7 A Second Course . 8 Acknowledgements . 9 0 Elementary Definitions 11 0.1 Rings and Ideals . 11 0.2 Unique Factorization . 13 0.3 Modules . 15 I Basic Constructions 19 1 Roots of Commutative Algebra 21 1.1 Number Theory . 21 1.2 Algebraic Curves and Function Theory . 23 1.3 Invariant Theory . 24 1.4 The Basis Theorem . 27 1.4.1 Finite Generation of Invariants . 29 viii Contents 1.5 Graded Rings . i0 1.6 Algebra and Geometry: The Nullstellensatz . 31 1.7 Geometric Invariant Theory . 37 1.8 Projective Varieties . 39 1.9 Hubert Functions and Polynomials . 42 1.10 Free Resolutions and the Syzygy Theorem . 44 1.11 Exercises . 46 Noetherian Rings and Modules . 4*> An Analysis of Hubert's Finiteness Argument . 47 Some Rings of Invariants . 48 Algebra and Geometry . 4^ Graded Rings and Projective Geometry . 52 Hubert Functions . 53 Free Resolutions . 54 Spec, max-Spec, and the Zariski Topology . 54 2 Localization 57 2.1 Fractions . 59 2.2 Hom and Tensor . 62 2.3 The Construction of Primes . 70 2.4 Rings and Modules of Finite Length . 71 2.5 Products of Domains . 78 2.6 Exercises . 78 Z-graded Rings and Their Localizations . 81 Partitions of Unity . 83 Gluing . 83 Constructing Primes . 84 Idempotents, Products, and Connected Components . 85 3 Associated Primes and Primary Decomposition 87 3.1 Associated Primes . 89 3.2 Prime Avoidance . 90 3.3 Primary Decomposition . 94 3.4 Primary Decomposition and Factoriality . 98 3.5 Primary Decomposition in the Graded Case . 99 3.6 Extracting Information from Primary Decomposition . . . 100 3.7 Why Primary Decomposition Is Not Unique . 102 3.8 Geometric Interpretation of Primary Decomposition . 103 3.9 Symbolic Powers and Functions Vanishing to High Order . 105 3.9.1 A Determinantal Example . 107 3.10 Exercises . 109 General Graded Primary Decomposition . 110 Primary Decomposition of Monomial Ideals . Ill The Question of Uniqueness . 112 Determinantal Ideals . 113 Contents ix Total Quotients. 113 Prime Avoidance . 114 4 Integral Dependence and the Nullstellensatz 117 4.1 The Cayley-Hamilton Theorem and Nakayama's Lemma . 119 4.2 Normal Domains and the Normalization Process . 125 4.3 Normalization in the Analytic Case . 128 4.4 Primes in an Integral Extension . 129 4.5 The Nullstellensatz. 131 4.6 Exercises . 135 Nakayama's Lemma . 136 Projective Modules and Locally Free Modules . 136 Integral Closure of Ideals . 137 Normalization . 138 Normalization and Convexity . 139 Nullstellensatz. 142 Three More Proofs of the Nullstellensatz. 142 5 Filtrations and the Artin-Rees Lemma 147 5.1 Associated Graded Rings and Modules . 148 5.2 The Blowup Algebra . 150 5.3 The Krull Intersection Theorem . 152 5.4 The Tangent Cone . 153 5.5 Exercises . 154 6 Flat Families 157 6.1 Elementary Examples . 159 6.2 Introduction to Tor . 161 6.3 Criteria for Flatness . 162 6.4 The Local Criterion for Flatness . 167 6.5 The Rees Algebra . 171 6.6 Exercises . 172 Flat Families of Graded Modules . 175 Embedded First-Order Deformations . 176 7 Completions and Hensel's Lemma 181 7.1 Examples and Definitions . 181 7.2 The Utility of Completions . 184 7.3 Lifting Idempotents . 188 7.4 Cohen Structure Theory and Coefficient Fields . 191 7.5 Basic Properties of Completion . 194 7.6 Maps from Power Series Rings . 200 7.7 Exercises . 205 Modules Whose Completions Are Isomorphic . 205 The Krull Topology and Cauchy Sequences . 206 Contents Completions from Power Series .207 Coefficient Fields . 207 Other Versions of Henseľs Lemma .208 II Dimension Theory 213 8 Introduction to Dimension Theory 215 8.1 Axioms for Dimension . 220 8.2 Other Characterizations of Dimension . 222 8.2.1 Affine Rings and Noether Normalization . 223 8.2.2 Systems of Parameters and Krull's Principal Ideal Theorem . 224 8.2.3 The Degree of the Hubert Polynomial . 225 9 Fundamental Definitions of Dimension Theory 227 9.1 Dimension Zero .229 9.2 Exercises .230 10 The Principal Ideal Theorem and Systems of Parameters 233 10.1 Systems of Parameters and Ideals of Finite Colength . . . 236 10.2 Dimension of Base and Fiber . 238 10.3 Regular Local Rings . 242 10.4 Exercises . 244 Determinantal Ideals .246 Hubert Series of a Graded Module .247 11 Dimension and Codimension One 251 11.1 Discrete Valuation Rings . 251 11.2 Normal Rings and Serre's Criterion . 253 11.3 Invertible Modules . 257 11.4 Unique Factorization of Codimension-One Ideals . 260 11.5 Divisors and Multiplicities . 262 11.6 Multiplicity of Principal Ideals . 265 11.7 Exercises . 268 Valuation Rings . 268 The Grothendieck Ring . 269 12 Dimension and Hilbert-Samuel Polynomials 275 12.1 Hilbert-Samuel Functions .276 12.2 Exercises .279 Analytic Spread and the Fiber of a Blowup .280 Multiplicities .280 Hubert Series .284 Contents xi 13 The Dimension of Affine Rings 285 13.1 Noether Normalization . 285 13.2 The Nullstellensatz. 296 13.3 Finiteness of the Integral Closure. 297 13.4 Exercises . 300 Quotients by Finite Groups . 300 Primes in Polynomial Rings . 301 Dimension in the Graded Case . 302 Noether Normalization in the Complete Case . 303 Products and Reduction to the Diagonal . 304 Equational Characterization of Systems of Parameters 306 14 Elimination Theory, Generic Freeness, and the Dimension of Fibers 307 14.1 Elimination Theory . 307 14.2 Generic Freeness . 312 14.3 The Dimension of Fibers . 313 14.4 Exercises . 318 Elimination Theory . 318 15 Gröbner Bases 321 Constructive Module Theory . 322 Elimination Theory . 322 15.1 Monomials and Terms . 323 151.1 Hubert Function and Polynomial . 324 15.1.2 Syzygies of Monomial Submodules . 326 15.2 Monomial Orders . 327 15.3 The Division Algorithm . 333 15.4 Gröbner Bases . 335 15.5 Syzygies . 337 15.6 History of Gröbner Bases . 340 15.7 A Property of Reverse Lexicographic Order . 342 15.8 Gröbner Bases and Flat Families . 345 15.9 Generic Initial Ideals . 351 15.9.1 Existence of the Generic Initial Ideal . 353 15.9.2 The Generic Initial Ideal is Borel-Fixed . 354 15.9.3 The Nature of Borel-Fixed Ideals . 355 15.10 Applications . 358 15.10.1 Ideal Membership . 359 15.10.2 Hubert Function and Polynomial . 359 15.10.3 Associated Graded Ring . 360 15.10.4 Elimination . 361 15.10.5 Projective Closure and Ideal at Infinity . 362 15.10.6 Saturation . 363 15.10.7 Lifting Homomorphisms . 364 xii Contents 15.10.8 Syzygies and Constructive Module Theory . 365 15.10.9 What's Left? . 367 15.11 Exercises . 368 15.12 Appendix: Some Computer Algebra Projects . 378 Project 1. Zero-dimensional Gorenstein Ideals . 376 Project 2. Factoring Out a General Element from an sth Syzygy . 377 Project 3. Resolutions over Hypersurfaces . 377 Project 4. Rational Curves of Degree Г+ 1 in Pr. . 378 Project 5. Regularity of Rational Curves . 378 Project 6. Some Monomial Curve Singularities . 379 Project 7. Some Interesting Prime Ideals . 379 16 Modules of Differentials 385 16.1 Computation of Differentials . 390 16.2 Differentials and the Cotangent Bundle . 390 16.3 Colimits and Localization . 393 16.4 Tangent Vector Fields and Infinitesimal Morphisms . 398 16.5 Differentials and Field Extensions . 400 16.6 Jacobian Criterion for Regularity . 404 16.7 Smoothness and Generic Smoothness . 407 16.8 Appendix: Another Construction of Kahler Differentials . 410 16.9 Exercises .412 III Homological Methods 421 17 Regular Sequences and the Koszul Complex 423 17.1 Koszul Complexes of Lengths 1 and 2. 424 17.2 Koszul Complexes in General . 427 17.3 Building the Koszul Complex from Parts . 431 17.4 Duality and Homotopies . 436 17.5 The Koszul Complex and the Cotangent Bundle of Projective Space . 440 17.6 Exercises . 441 Free Resolutions of Monomial Ideals . 443 Conormal Sequence of a Complete Intersection . 444 Regular Sequences Are Like Sequences of Variables . 445 Blowup Algebra and Normal Cone of a Regular Sequence . 445 Geometric Contexts of the Koszul Complex .447 Contents xiii 18 Depth, Codimension, and Cohen-Macaulay Rings 451 18.1 Depth . 451 18.1.1 Depth and the Vanishing of Ext . 453 18.2 Cohen-Macaulay Rings . 455 18.3 Proving Primeness with Serre's Criterion . 461 18.4 Flatness and Depth . 464 18.5 Some Examples . 466 18.6 Exercises . 469 19 Homological Theory of Regular Local Rings 473 19.1 Projective Dimension and Minimal Resolutions . 473 19.2 Global Dimension and the Syzygy Theorem . 478 19.3 Depth and Projective Dimension: The Auslander- Buchsbaum Formula . 479 19.4 Stably Free Modules and Factoriality of Regular Local Rings . 484 19.5 Exercises . 488 Regular Rings . 488 Modules over a Dedekind Domain . 488 The Auslander-Buchsbaum Formula . 489 Projective Dimension and Cohen-Macaulay Rings . . 489 Hubert Function and Grothendieck Group . 490 The Chern Polynomial . 492 20 Free Resolutions and Fitting Invariants 493 20.1 The Uniqueness of Free Resolutions . 494 20.2 Fitting Ideals . 496 20.3 What Makes a Complex Exact? . 500 20.4 The Hilbert-Burch Theorem . 506 20.4.1 Cubic Surfaces and Sextuples of Points in the Plane . 508 20.5 Castelnuovo-Mumford Regularity . 509 20.5.1 Regularity and Hyperplane Sections . 513 20.5.2 Regularity of Generic Initial Ideals . 514 20.5.3 Historical Notes on Regularity . 514 20.6 Exercises . 515 Fitting Ideals and the Structure of Modules . 515 Projectives of Constant Rank . 518 Castelnuovo-Mumford Regularity . 521 21 Duality, Canonical Modules, and Gorenstein Rings 523 21.1 Duality for Modules of Finite Length . 524 21.2 Zero-Dimensional Gorenstein Rings . 529 21.3 Canonical Modules and Gorenstein Rings in Higher Dimension . 532 xiv Contents 21.4 Maximal Cohen-Macaulay Modules. 533 21.5 Modules of Finite Injective Dimension . 534 21.6 Uniqueness and (Often) Existence . 538 21.7 Localization and Completion of the Canonical Module . . 540 21.8 Complete Intersections and Other Gorenstein Rings . 541 21.9 Duality for Maximal Cohen-Macaulay Modules . 542 21.10 Linkage .-543 21.11 Duality in the Graded Case . 549 21.12 Exercises . 550 The Zero-Dimensional Case and Duality . 550 Higher Dimension . 552 The Canonical Module as Ideal . 555 Linkage and the Cayley-Bacharach Theorem . 556 Appendix 1 Field Theory 555 Al.l Transcendence Degree .561 A1.2 Separability .563 A1.3 p-Bases .565 Al.3.1 Exercises .568 Appendix 2 Multilinear Algebra 565 A2.1 Introduction . 571 A2.2 Tensor Products . 573 A2.3 Symmetric and Exterior Algebras . 574 A2.3.1 Bases . 578 A2.3.2 Exercises . 580 A2.4 Coalgebra Structures and Divided Powers . 581 A2.4.1 S(M)· and S(M) as Modules over One Another . 582 A2.5 Schur Functors . 590 A2.5.1 Exercises . 594 A2.6 Complexes Constructed by Multilinear Algebra . 596 A2.6.1 Strands of the Koszul Complex . 597 A2.6.2 Exercises . 609 Appendix 3 Homological Algebra 611 A3.1 Introduction . 617 Part I: Resolutions and Derived Functors . 614 A3.2 Free and Projective Modules . 621 A3.3 Free and Projective Resolutions . 623 A3.4 Injective Modules and Resolutions . 624 A3.4.1 Exercises . 630 Injective Envelopes . 630 Injective Modules over Noetherian Rings . 630 A3.5 Basic Constructions with Complexes . 632 A3.5.1 Notation and Definitions . 632 Contents xv A3.6 Maps and Homotopies of Complexes . 633 A3.7 Exact Sequences of Complexes . 637 АЗ^Л Exercises . 638 A3.8 The Long Exact Sequence in Homology . 639 A3.8.1 Exercises . 640 Diagrams and Syzygies . 640 A3.9 Derived Functors . 643 A3.9.1 Exercise on Derived Functors . 645 АЗ.ЮТог . 646 A3.10.1 Exercises: Tor . 646 A3.ll Ext . 649 АЗ.П.І Exercises: Ext . 651 A3.11.2 Local Cohomology . 656 Part II: From Mapping Cones to Spectral Sequences . . 650 A3.12 The Mapping Cone and Double Complexes . 656 A3.12.1 Exercises: Mapping Cones and Double Complexes 660 A3.13 Spectral Sequences . 663 A3.13.1 Mapping Cones Revisited . 664 A3.13.2 Exact Couples . 665 A3.13.3 Filtered Differential Modules and Complexes . 668 A3.13.4 The Spectral Sequence of a Double Complex . 671 A3.13.5 Exact Sequence of Terms of Low Degree . 677 A3.13.6 Exercises on Spectral Sequences . 678 A3.14 Derived Categories . 684 A3.14.1 Step One: The Homotopy Category of Complexes 685 A3.14.2 Step Two: The Derived Category . 686 A3.14.3 Exercises on the Derived Category . 688 Appendix 4 Λ Sketch of Local Cohomology 683 A4.1 Local Cohomology and Global Cohomology . 693 A4.2 Local Duality . 694 A4.3 Depth and Dimension . 695 Appendix 5 Category Theory 689 A5.1 Categories, Functors, and Natural Transformations . 697 A5.2 Adjoint Functors . 699 A5.2.1 Uniqueness . 700 A5.2.2 Some Examples . 700 A5.2.3 Another Characterization of Adjoints . 701 A5.2.4 Adjoints and Limits . 702 A5.3 Representable Functors and Yoneda's Lemma . 703 xvi Contents Appendix 6 Limits and Colimits 697 A6.1 Colimits in the Category of Modules.708 A6.2 Flat Modules as Colimits of Free Modules.711 A6.3 Colimits in the Category of Commutative Algebras . 713 A6.4 Exercises .715 Appendix 7 Where Next? 709 Hints and Solutions for Selected1 Exercises 711 References 757 Index of Notation 775 Index 779
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spelling	Eisenbud, David 1947- Verfasser (DE-588)139999671 aut Commutative algebra with a view toward algebraic geometry David Eisenbud [7. print] New York [u.a.] Springer 2004 XVI, 797 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 150 Literaturverz. S. 757 - 774 Anéis e álgebras comutativos larpcal Geometria algébrica larpcal Commutative algebra Geometry, Algebraic Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Kommutative Algebra (DE-588)4164821-3 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 s Kommutative Algebra (DE-588)4164821-3 s 1\p DE-604 DE-604 Graduate texts in mathematics 150 (DE-604)BV000000067 150 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014574789&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
spellingShingle	Eisenbud, David 1947- Commutative algebra with a view toward algebraic geometry Graduate texts in mathematics Anéis e álgebras comutativos larpcal Geometria algébrica larpcal Commutative algebra Geometry, Algebraic Algebraische Geometrie (DE-588)4001161-6 gnd Kommutative Algebra (DE-588)4164821-3 gnd
subject_GND	(DE-588)4001161-6 (DE-588)4164821-3
title	Commutative algebra with a view toward algebraic geometry
title_auth	Commutative algebra with a view toward algebraic geometry
title_exact_search	Commutative algebra with a view toward algebraic geometry
title_exact_search_txtP	Commutative algebra with a view toward algebraic geometry
title_full	Commutative algebra with a view toward algebraic geometry David Eisenbud
title_fullStr	Commutative algebra with a view toward algebraic geometry David Eisenbud
title_full_unstemmed	Commutative algebra with a view toward algebraic geometry David Eisenbud
title_short	Commutative algebra with a view toward algebraic geometry
title_sort	commutative algebra with a view toward algebraic geometry
topic	Anéis e álgebras comutativos larpcal Geometria algébrica larpcal Commutative algebra Geometry, Algebraic Algebraische Geometrie (DE-588)4001161-6 gnd Kommutative Algebra (DE-588)4164821-3 gnd
topic_facet	Anéis e álgebras comutativos Geometria algébrica Commutative algebra Geometry, Algebraic Algebraische Geometrie Kommutative Algebra
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work_keys_str_mv	AT eisenbuddavid commutativealgebrawithaviewtowardalgebraicgeometry

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