The rising sea: Foundations of Algebraic Geometry

801 Pages·2015·3.44 MB·english

by  Ravi Vakil

THE RISING SEA
Foundations of Algebraic Geometry
math216.wordpress.com
December29,2015draft
⃝c 2010–2015byRaviVakil.
Notetoreader:thefigures,index,andformattinghaveyettobeproperlydealtwith.
Thereremainmanyissuesstilltobedealtwithinthemainpartofthenotes(including
manyofyourcorrectionsandsuggestions).

Contents
Preface 11
0.1. Forthereader 12
0.2. Fortheexpert 16
0.3. Backgroundandconventions 17
0.4. ⋆⋆Thegoalsofthisbook 18
PartI. Preliminaries 21
Chapter1. Somecategorytheory 23
1.1. Motivation 23
1.2. Categoriesandfunctors 25
1.3. Universalpropertiesdetermineanobjectuptouniqueisomorphism 31
1.4. Limitsandcolimits 40
1.5. Adjoints 44
1.6. Anintroductiontoabeliancategories 47
1.7. ⋆Spectralsequences 58
Chapter2. Sheaves 71
2.1. Motivatingexample: Thesheafofdifferentiablefunctions 71
2.2. Definitionofsheafandpresheaf 73
2.3. Morphismsofpresheavesandsheaves 78
2.4. Propertiesdeterminedatthelevelofstalks,andsheafification 82
2.5. Sheavesofabeliangroups,andO -modules,formabeliancategories 86
X
2.6. Theinverseimagesheaf 89
2.7. Recoveringsheavesfroma“sheafonabase” 92
PartII. Schemes 97
Chapter3. Towardaffineschemes: theunderlyingset,andtopologicalspace 99
3.1. Towardschemes 99
3.2. Theunderlyingsetofaffineschemes 101
3.3. VisualizingschemesI:genericpoints 113
3.4. Theunderlyingtopologicalspaceofanaffinescheme 114
3.5. AbaseoftheZariskitopologyonSpecA: Distinguishedopensets 117
3.6. Topological(andNoetherian)properties 118
3.7. ThefunctionI((cid:1)),takingsubsetsofSpecAtoidealsofA 127
Chapter4. Thestructuresheaf,andthedefinitionofschemesingeneral 129
4.1. Thestructuresheafofanaffinescheme 129
4.2. VisualizingschemesII:nilpotents 133
3
4.3. Definitionofschemes 136
4.4. Threeexamples 139
4.5. Projectiveschemes,andtheProjconstruction 145
Chapter5. Somepropertiesofschemes 153
5.1. Topologicalproperties 153
5.2. Reducednessandintegrality 155
5.3. Propertiesofschemesthatcanbechecked“affine-locally” 157
5.4. Normalityandfactoriality 161
5.5. Thecrucialpointsofaschemethatcontroleverything: Associated
pointsandprimes 166
PartIII. Morphisms 175
Chapter6. Morphismsofschemes 177
6.1. Introduction 177
6.2. Morphismsofringedspaces 178
6.3. Fromlocallyringedspacestomorphismsofschemes 180
6.4. Mapsofgradedringsandmapsofprojectiveschemes 186
6.5. Rationalmapsfromreducedschemes 188
6.6. ⋆Representablefunctorsandgroupschemes 194
6.7. ⋆⋆TheGrassmannian(initialconstruction) 199
Chapter7. Usefulclassesofmorphismsofschemes 201
7.1. Anexampleofareasonableclassofmorphisms: Openembeddings 201
7.2. Algebraicinterlude: LyingOverandNakayama 203
7.3. Agazillionfinitenessconditionsonmorphisms 207
7.4. Imagesofmorphisms: Chevalley’sTheoremandeliminationtheory 216
Chapter8. Closedembeddingsandrelatednotions 225
8.1. Closedembeddingsandclosedsubschemes 225
8.2. Moreprojectivegeometry 230
8.3. The(closedsub)scheme-theoreticimage 236
8.4. EffectiveCartierdivisors,regularsequencesandregularembeddings240
Chapter9. Fiberedproductsofschemes,andbasechange 245
9.1. Theyexist 245
9.2. Computingfiberedproductsinpractice 251
9.3. Interpretations: Pullingbackfamilies,andfibersofmorphisms 254
9.4. Propertiespreservedbybasechange 260
9.5. ⋆Propertiesnotpreservedbybasechange,andhowtofixthem 261
9.6. Productsofprojectiveschemes: TheSegreembedding 269
9.7. Normalization 271
Chapter10. Separatedandpropermorphisms,and(finally!) varieties 277
10.1. Separatedmorphisms(andquasiseparatednessdoneproperly) 277
10.2. Rationalmapstoseparatedschemes 287
10.3. Propermorphisms 291
PartIV. “Geometric”properties: Dimensionandsmoothness 299
Chapter11. Dimension 301
11.1. Dimensionandcodimension 301
11.2. Dimension,transcendencedegree,andNoethernormalization 305
11.3. Codimensiononemiracles: Krull’sandHartogs’sTheorems 313
11.4. Dimensionsoffibersofmorphismsofvarieties 319
11.5. ⋆⋆ProofofKrull’sPrincipalIdealandHeightTheorems 324
Chapter12. Regularityandsmoothness 327
12.1. TheZariskitangentspace 327
12.2. Regularity,andsmoothnessoverafield 333
12.3. Examples 338
12.4. Bertini’sTheorem 341
12.5. Discretevaluationrings 345
12.6. Smooth(ande´tale)morphisms(firstdefinition) 350
12.7. ⋆Valuativecriteriaforseparatednessandproperness 353
12.8. ⋆Moresophisticatedfactsaboutregularlocalrings 358
12.9. ⋆Filteredringsandmodules,andtheArtin-ReesLemma 359
PartV. Quasicoherentsheaves 363
Chapter13. Quasicoherentandcoherentsheaves 365
13.1. Vectorbundlesandlocallyfreesheaves 365
13.2. Quasicoherentsheaves 371
13.3. Characterizingquasicoherenceusingthedistinguishedaffinebase 373
13.4. Quasicoherentsheavesformanabeliancategory 377
13.5. Module-likeconstructions 379
13.6. Finitetypeandcoherentsheaves 382
13.7. Pleasantpropertiesoffinitetypeandcoherentsheaves 385
13.8. ⋆⋆Coherentmodulesovernon-Noetherianrings 389
Chapter14. Linebundles: Invertiblesheavesanddivisors 393
14.1. Somelinebundlesonprojectivespace 393
14.2. LinebundlesandWeildivisors 395
14.3. ⋆EffectiveCartierdivisors“=”invertibleidealsheaves 404
Chapter15. QuasicoherentsheavesonprojectiveA-schemes 407
15.1. Thequasicoherentsheafcorrespondingtoagradedmodule 407
15.2. Invertiblesheaves(linebundles)onprojectiveA-schemes 408
15.3. Globallygeneratedandbase-point-freelinebundles 409
15.4. Quasicoherentsheavesandgradedmodules 412
Chapter16. Pushforwardsandpullbacksofquasicoherentsheaves 417
16.1. Introduction 417
16.2. Pushforwardsofquasicoherentsheaves 417
16.3. Pullbacksofquasicoherentsheaves 418
16.4. Linebundlesandmapstoprojectiveschemes 424
16.5. TheCurve-to-ProjectiveExtensionTheorem 431
16.6. Ampleandveryamplelinebundles 432
16.7. ⋆TheGrassmannianasamodulispace 437
Chapter17. RelativeversionsofSpecandProj,andprojectivemorphisms 443
17.1. RelativeSpecofa(quasicoherent)sheafofalgebras 443
17.2. RelativeProjofasheafofgradedalgebras 446
17.3. Projectivemorphisms 449
17.4. Applicationstocurves 455
Chapter18. Cˇechcohomologyofquasicoherentsheaves 461
18.1. (Desired)propertiesofcohomology 461
18.2. Definitionsandproofsofkeyproperties 466
18.3. Cohomologyoflinebundlesonprojectivespace 471
18.4. Riemann-Roch,degreesofcoherentsheaves,andarithmeticgenus 474
18.5. AfirstglimpseofSerreduality 481
18.6. Hilbertfunctions,Hilbertpolynomials,andgenus 485
18.7. ⋆Serre’scohomologicalcharacterizationofampleness 490
18.8. Higherpushforward(ordirectimage)sheaves 493
18.9. ⋆ From projective to proper hypotheses: Chow’s Lemma and
Grothendieck’sCoherenceTheorem 497
Chapter19. Application: Curves 503
19.1. Acriterionforamorphismtobeaclosedembedding 503
19.2. Aseriesofcrucialtools 505
19.3. Curvesofgenus0 508
19.4. Classicalgeometryarisingfromcurvesofpositivegenus 509
19.5. Hyperellipticcurves 511
19.6. Curvesofgenus2 515
19.7. Curvesofgenus3 516
19.8. Curvesofgenus4and5 518
19.9. Curvesofgenus1 521
19.10. Ellipticcurvesaregroupvarieties 528
19.11. Counterexamplesandpathologiesusingellipticcurves 534
Chapter20. ⋆Application: Aglimpseofintersectiontheory 539
20.1. Intersectingnlinebundleswithann-dimensionalvariety 539
20.2. Intersectiontheoryonasurface 543
20.3. The Grothendieck group of coherent sheaves, and an algebraic
versionofhomology 549
20.4. ⋆⋆TheNakai-MoishezonandKleimancriteriaforampleness 551
Chapter21. Differentials 557
21.1. Motivationandgameplan 557
21.2. Definitionsandfirstproperties 558
21.3. Smoothnessofvarietiesrevisited 572
21.4. Examples 575
21.5. Understandingsmoothvarietiesusingtheircotangentbundles 579
21.6. Unramifiedmorphisms 584
21.7. TheRiemann-HurwitzFormula 586
Chapter22. ⋆Blowingup 593
22.1. Motivatingexample: blowinguptheoriginintheplane 593
22.2. Blowingup,byuniversalproperty 595
22.3. Theblow-upexists,andisprojective 599
22.4. Examplesandcomputations 604
PartVI. More 613
Chapter23. Derivedfunctors 615
23.1. TheTorfunctors 615
23.2. Derivedfunctorsingeneral 619
23.3. Derivedfunctorsandspectralsequences 623
23.4. DerivedfunctorcohomologyofO-modules 628
23.5. Cˇechcohomologyandderivedfunctorcohomologyagree 631
Chapter24. Flatness 639
24.1. Introduction 639
24.2. Easierfacts 641
24.3. FlatnessthroughTor 646
24.4. Ideal-theoreticcriteriaforflatness 648
24.5. Topologicalaspectsofflatness 655
24.6. Localcriteriaforflatness 659
24.7. FlatnessimpliesconstantEulercharacteristic 663
Chapter25. Smoothande´talemorphisms,andflatness 669
25.1. Somemotivation 669
25.2. Differentcharacterizationsofsmoothande´talemorphisms 671
25.3. GenericsmoothnessandtheKleiman-BertiniTheorem 676
Chapter26. DepthandCohen-Macaulayness 681
26.1. Depth 681
26.2. Cohen-Macaulayringsandschemes 684
26.3. ⋆Serre’sR1+S2criterionfornormality 687
Chapter27. Twenty-sevenlines 693
27.1. Introduction 693
27.2. Preliminaryfacts 694
27.3. Everysmoothcubicsurface(overk)has27lines 695
27.4. Everysmoothcubicsurface(overk)isablownupplane 699
Chapter28. Cohomologyandbasechangetheorems 703
28.1. Statementsandapplications 703
28.2. ⋆Proofsofcohomologyandbasechangetheorems 709
28.3. Applyingcohomologyandbasechangetomoduliproblems 716
Chapter29. PowerseriesandtheTheoremonFormalFunctions 721
29.1. Introduction 721
29.2. Algebraicpreliminaries 721
29.3. Definingtypesofsingularities 725
29.4. TheTheoremonFormalFunctions 727
29.5. Zariski’sConnectednessLemmaandSteinFactorization 729
29.6. Zariski’sMainTheorem 732
29.7. Castelnuovo’sCriterionforcontracting(-1)-curves 735
29.8. ⋆⋆ProofoftheTheoremonFormalFunctions29.4.2 739
Chapter30. ⋆ProofofSerreduality 743
30.1. Introduction 743
30.2. ExtgroupsandExtsheavesforO-modules 748
30.3. Serredualityforprojectivek-schemes 752
30.4. Theadjunctionformulaforthedualizingsheaf,and! =K 757
X X
Bibliography 763
Index 769
December29,2015draft 9
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