The Stacks Project: Algebraic Stacks and the Algebraic Geometry Needed to Define Them

5338 Pages·2016·25 MB·English

by  The Stacks Project

Stacks Project
Version 1a50e77, compiled on Jun 30, 2016.
Thefollowingpeoplehavecontributedtothiswork: KianAbolfazlian,DanAbramovich,
Juan Pablo Acosta Lopez, Shishir Agrawal, Jarod Alper, Dima Arinkin, Aravind
Asok, Giulia Battiston, Hanno Becker, Mark Behrens, Pieter Belmans, Olivier
Benoist, Daniel Bergh, Michel Van den Bergh, Bhargav Bhatt, Wessel Bindt, Ingo
Blechschmidt, Lucas Braune, Martin Bright, David Brown, Niels Borne, Ragnar-
Olaf Buchweitz, Robert Cardona, Nuno Cardoso, Scott Carnahan, Kestutis Ces-
navicius, Antoine Chambert-Loir, Will Chen, Filip Chindea, Nava Chitrik, Fraser
Chiu, Dustin Clausen, J´er´emy Cochoy, Johan Commelin, Brian Conrad, David
Corwin, Peadar Coyle, Rankeya Datta, Aise Johan de Jong, Matt DeLand, Ash-
winDeopurkar, MaartenDerickx, BenjaminDiamond, DanielDisegni, JoelDodge,
Taylor Dupuy, Bas Edixhoven, Alexander Palen Ellis, Matthew Emerton, Andrew
Fanoe, Maxim Fedorchuck, Hu Fei, Dan Fox, Cameron Franc, Dragos Fratila,
RobertFriedman,OferGabber,LennartGalinat,MartinGallauer,LuisGarcia,Xu
Gao, Toby Gee, Anton Geraschenko, Daniel Gerigk, Alberto Gioia, Julia Ramos
Gonzalez, Jean-Pierre Gourdot, Darij Grinberg, Yuzhou Gu, Zeshen Gu, Quentin
Guignard, Albert Gunawan, Joseph Gunther, Andrei Halanay, Yatir Halevi, Jack
Hall,DanielHalpern-Leistner,XueHang,DavidHansen,YunHao,MichaelHarris,
William Hart, Philipp Hartwig, Mohamed Hashi, Olivier Haution, Florian Hei-
derich, Jeremiah Heller, Kristen Hendricks, Fraser Hiu, Quoc P. Ho, Amit Hogadi,
David Holmes, Andreas Holmstrom, Ray Hoobler, John Hosack, Xiaowen Hu,
Yuhao Huang, Yu-Liang Huang, Ariyan Javanpeykar, Lena Min Ji, Peter Johnson,
Christian Kappen, Kiran Kedlaya, Timo Keller, Adeel Ahmad Khan, Keenan Kid-
well,AndrewKiluk,LarsKindler,J´anosKoll´ar,S´andorKov´acs,EmmanuelKowal-
ski, Dmitry Korb, Girish Kulkarni, Matthias Kummerer, Daniel Krashen, Geoffrey
Lee, Min Lee, Simon Pepin Lehalleur, Tobi Lehman, Florian Lengyel, Pak-Hin
Lee, Brandon Levin, Paul Lessard, Mao Li, Shizhang Li, Max Lieblich, Hsing Liu,
Qing Liu, David Lubicz, Zachary Maddock, Mohammed Mammeri, Sonja Mapes,
Florent Martin, Akhil Mathew, Daniel Miller, Yogesh More, Laurent Moret-Bailly,
Maxim Mornev, Jackson Morrow, Yusuf Mustopa, David Mykytyn, Josh Nichols-
Barrer, Kien Nguyen, Thomas Nyberg, Masahiro Ohno, Catherine O’Neil, Martin
Olsson, Brian Osserman, Thanos Papaioannou, Roland Paulin, Rakesh Pawar, Pe-
ter Percival, Alex Perry, Gregor Pohl, Bjorn Poonen, Anatoly Preygel, Artem Pri-
hodko,ThibautPugin,YouQi,RyanReich,CharlesRezk,AliceRizzardo,Herman
Rohrbach,FredRohrer,MatthieuRomagny,JoeRoss,JuliusRoss,ApurbaKumar
Roy, Rob Roy, David Rydh, Jyoti Prakash Saha, Beren Sanders, Olaf Schnu¨rer,
Jakob Scholbach, Rene Schoof, Jaakko Seppala, Michele Serra, Chung-chieh Shan,
Liran Shaul, Minseon Shin, Jeroen Sijsling, Thomas Smith, Tanya Kaushal Sri-
vastava, Axel St¨abler, Jason Starr, Thierry Stulemeijer, Takashi Suzuki, Lenny
Taelman, Abolfazl Tarizadeh, John Tate, Titus Teodorescu, Michael Thaddeus,
Stulemeijer Thierry, Shabalin Timofey, Alex Torzewski, Burt Totaro, Ravi Vakil,
Theo van den Bogaart, Remy van Dobben de Bruyn, Kevin Ventullo, Hendrik
Verhoek, Erik Visse, Angelo Vistoli, Konrad Voelkel, Rishi Vyas, James Waldron,
Hua Wang, Jonathan Wang, Matthew Ward, Evan Warner, John Watterlond, Ian
Whitehead, Jonathan Wise, William Wright, Wei Xu, Qijun Yan, Amnon Yeku-
tieli, Alex Youcis, John Yu, Felipe Zaldivar, Zhe Zhang, Yifei Zhao, Yu Zhao, Fan
Zheng,WeizheZheng,AnfangZhou,FanZhou,WouterZomervrucht,RunpuZong,
Jeroen Zuiddam, David Zureick-Brown.
3
Copyright (C) 2005 -- 2016 Johan de Jong
Permission is granted to copy, distribute and/or modify this
document under the terms of the GNU Free Documentation License,
Version 1.2 or any later version published by the Free Software
Foundation; with no Invariant Sections, no Front-Cover Texts,
and no Back-Cover Texts. A copy of the license is included in
the section entitled "GNU Free Documentation License".
Contents
Chapter 1. Introduction 61
1.1. Overview 61
1.2. Attribution 61
1.3. Other chapters 62
Chapter 2. Conventions 64
2.1. Comments 64
2.2. Set theory 64
2.3. Categories 64
2.4. Algebra 64
2.5. Notation 64
2.6. Other chapters 64
Chapter 3. Set Theory 66
3.1. Introduction 66
3.2. Everything is a set 66
3.3. Classes 66
3.4. Ordinals 67
3.5. The hierarchy of sets 67
3.6. Cardinality 67
3.7. Cofinality 68
3.8. Reflection principle 68
3.9. Constructing categories of schemes 69
3.10. Sets with group action 74
3.11. Coverings of a site 75
3.12. Abelian categories and injectives 77
3.13. Other chapters 77
Chapter 4. Categories 79
4.1. Introduction 79
4.2. Definitions 79
4.3. Opposite Categories and the Yoneda Lemma 83
4.4. Products of pairs 84
4.5. Coproducts of pairs 85
4.6. Fibre products 85
4.7. Examples of fibre products 87
4.8. Fibre products and representability 87
4.9. Pushouts 88
4.10. Equalizers 89
4.11. Coequalizers 89
4
CONTENTS 5
4.12. Initial and final objects 90
4.13. Monomorphisms and Epimorphisms 90
4.14. Limits and colimits 90
4.15. Limits and colimits in the category of sets 93
4.16. Connected limits 93
4.17. Cofinal and initial categories 94
4.18. Finite limits and colimits 96
4.19. Filtered colimits 98
4.20. Cofiltered limits 102
4.21. Limits and colimits over partially ordered sets 102
4.22. Essentially constant systems 105
4.23. Exact functors 108
4.24. Adjoint functors 108
4.25. A criterion for representability 110
4.26. Localization in categories 112
4.27. Formal properties 123
4.28. 2-categories 126
4.29. (2, 1)-categories 128
4.30. 2-fibre products 128
4.31. Categories over categories 134
4.32. Fibred categories 136
4.33. Inertia 142
4.34. Categories fibred in groupoids 144
4.35. Presheaves of categories 150
4.36. Presheaves of groupoids 152
4.37. Categories fibred in sets 153
4.38. Categories fibred in setoids 155
4.39. Representable categories fibred in groupoids 157
4.40. Representable 1-morphisms 158
4.41. Other chapters 161
Chapter 5. Topology 163
5.1. Introduction 163
5.2. Basic notions 163
5.3. Hausdorff spaces 164
5.4. Bases 164
5.5. Submersive maps 165
5.6. Connected components 166
5.7. Irreducible components 168
5.8. Noetherian topological spaces 172
5.9. Krull dimension 173
5.10. Codimension and catenary spaces 174
5.11. Quasi-compact spaces and maps 175
5.12. Locally quasi-compact spaces 178
5.13. Limits of spaces 181
5.14. Constructible sets 183
5.15. Constructible sets and Noetherian spaces 186
5.16. Characterizing proper maps 187
5.17. Jacobson spaces 190
CONTENTS 6
5.18. Specialization 192
5.19. Dimension functions 194
5.20. Nowhere dense sets 196
5.21. Profinite spaces 197
5.22. Spectral spaces 198
5.23. Limits of spectral spaces 203
5.24. Stone-Cˇech compactification 206
5.25. Extremally disconnected spaces 207
5.26. Miscellany 210
5.27. Partitions and stratifications 211
5.28. Colimits of spaces 212
5.29. Topological groups, rings, modules 213
5.30. Other chapters 216
Chapter 6. Sheaves on Spaces 218
6.1. Introduction 218
6.2. Basic notions 218
6.3. Presheaves 218
6.4. Abelian presheaves 219
6.5. Presheaves of algebraic structures 220
6.6. Presheaves of modules 221
6.7. Sheaves 222
6.8. Abelian sheaves 224
6.9. Sheaves of algebraic structures 224
6.10. Sheaves of modules 226
6.11. Stalks 226
6.12. Stalks of abelian presheaves 227
6.13. Stalks of presheaves of algebraic structures 228
6.14. Stalks of presheaves of modules 228
6.15. Algebraic structures 229
6.16. Exactness and points 230
6.17. Sheafification 231
6.18. Sheafification of abelian presheaves 233
6.19. Sheafification of presheaves of algebraic structures 234
6.20. Sheafification of presheaves of modules 235
6.21. Continuous maps and sheaves 236
6.22. Continuous maps and abelian sheaves 240
6.23. Continuous maps and sheaves of algebraic structures 241
6.24. Continuous maps and sheaves of modules 243
6.25. Ringed spaces 246
6.26. Morphisms of ringed spaces and modules 246
6.27. Skyscraper sheaves and stalks 248
6.28. Limits and colimits of presheaves 249
6.29. Limits and colimits of sheaves 249
6.30. Bases and sheaves 252
6.31. Open immersions and (pre)sheaves 259
6.32. Closed immersions and (pre)sheaves 264
6.33. Glueing sheaves 265
6.34. Other chapters 267
CONTENTS 7
Chapter 7. Sites and Sheaves 269
7.1. Introduction 269
7.2. Presheaves 269
7.3. Injective and surjective maps of presheaves 270
7.4. Limits and colimits of presheaves 271
7.5. Functoriality of categories of presheaves 271
7.6. Sites 274
7.7. Sheaves 275
7.8. Families of morphisms with fixed target 277
7.9. The example of G-sets 280
7.10. Sheafification 282
7.11. Quasi-compact objects and colimits 287
7.12. Injective and surjective maps of sheaves 290
7.13. Representable sheaves 291
7.14. Continuous functors 292
7.15. Morphisms of sites 294
7.16. Topoi 295
7.17. G-sets and morphisms 297
7.18. More functoriality of presheaves 298
7.19. Cocontinuous functors 300
7.20. Cocontinuous functors and morphisms of topoi 302
7.21. Cocontinuous functors which have a right adjoint 306
7.22. Cocontinuous functors which have a left adjoint 306
7.23. Existence of lower shriek 307
7.24. Localization 308
7.25. Glueing sheaves 311
7.26. More localization 313
7.27. Localization and morphisms 314
7.28. Morphisms of topoi 318
7.29. Localization of topoi 323
7.30. Localization and morphisms of topoi 326
7.31. Points 328
7.32. Constructing points 332
7.33. Points and morphisms of topoi 334
7.34. Localization and points 336
7.35. 2-morphisms of topoi 338
7.36. Morphisms between points 339
7.37. Sites with enough points 339
7.38. Criterion for existence of points 341
7.39. Weakly contractible objects 343
7.40. Exactness properties of pushforward 344
7.41. Almost cocontinuous functors 348
7.42. Subtopoi 350
7.43. Sheaves of algebraic structures 352
7.44. Pullback maps 355
7.45. Topologies 356
7.46. The topology defined by a site 359
7.47. Sheafification in a topology 361
CONTENTS 8
7.48. Topologies and sheaves 364
7.49. Topologies and continuous functors 365
7.50. Points and topologies 365
7.51. Other chapters 365
Chapter 8. Stacks 367
8.1. Introduction 367
8.2. Presheaves of morphisms associated to fibred categories 367
8.3. Descent data in fibred categories 369
8.4. Stacks 371
8.5. Stacks in groupoids 375
8.6. Stacks in setoids 376
8.7. The inertia stack 379
8.8. Stackification of fibred categories 379
8.9. Stackification of categories fibred in groupoids 383
8.10. Inherited topologies 384
8.11. Gerbes 387
8.12. Functoriality for stacks 391
8.13. Stacks and localization 399
8.14. Other chapters 400
Chapter 9. Fields 402
9.1. Introduction 402
9.2. Basic definitions 402
9.3. Examples of fields 402
9.4. Vector spaces 403
9.5. The characteristic of a field 404
9.6. Field extensions 404
9.7. Finite extensions 406
9.8. Algebraic extensions 408
9.9. Minimal polynomials 410
9.10. Algebraic closure 411
9.11. Relatively prime polynomials 413
9.12. Separable extensions 413
9.13. Linear independence of characters 417
9.14. Purely inseparable extensions 418
9.15. Normal extensions 420
9.16. Splitting fields 422
9.17. Roots of unity 424
9.18. Finite fields 424
9.19. Primitive elements 424
9.20. Trace and norm 425
9.21. Galois theory 428
9.22. Infinite Galois theory 430
9.23. The complex numbers 433
9.24. Kummer extensions 434
9.25. Artin-Schreier extensions 434
9.26. Transcendence 434
9.27. Linearly disjoint extensions 437
CONTENTS 9
9.28. Review 438
9.29. Other chapters 439
Chapter 10. Commutative Algebra 441
10.1. Introduction 441
10.2. Conventions 441
10.3. Basic notions 441
10.4. Snake lemma 443
10.5. Finite modules and finitely presented modules 444
10.6. Ring maps of finite type and of finite presentation 446
10.7. Finite ring maps 447
10.8. Colimits 447
10.9. Localization 451
10.10. Internal Hom 456
10.11. Tensor products 457
10.12. Tensor algebra 461
10.13. Base change 463
10.14. Miscellany 465
10.15. Cayley-Hamilton 466
10.16. The spectrum of a ring 468
10.17. Local rings 472
10.18. The Jacobson radical of a ring 473
10.19. Nakayama’s lemma 474
10.20. Open and closed subsets of spectra 475
10.21. Connected components of spectra 476
10.22. Glueing functions 477
10.23. More glueing results 480
10.24. Zerodivisors and total rings of fractions 483
10.25. Irreducible components of spectra 484
10.26. Examples of spectra of rings 485
10.27. A meta-observation about prime ideals 489
10.28. Images of ring maps of finite presentation 491
10.29. More on images 494
10.30. Noetherian rings 496
10.31. Locally nilpotent ideals 498
10.32. Curiosity 500
10.33. Hilbert Nullstellensatz 501
10.34. Jacobson rings 502
10.35. Finite and integral ring extensions 510
10.36. Normal rings 514
10.37. Going down for integral over normal 518
10.38. Flat modules and flat ring maps 520
10.39. Supports and annihilators 526
10.40. Going up and going down 528
10.41. Separable extensions 531
10.42. Geometrically reduced algebras 533
10.43. Separable extensions, continued 535
10.44. Perfect fields 537
10.45. Universal homeomorphisms 538
CONTENTS 10
10.46. Geometrically irreducible algebras 542
10.47. Geometrically connected algebras 545
10.48. Geometrically integral algebras 547
10.49. Valuation rings 547
10.50. More Noetherian rings 551
10.51. Length 552
10.52. Artinian rings 556
10.53. Homomorphisms essentially of finite type 557
10.54. K-groups 558
10.55. Graded rings 561
10.56. Proj of a graded ring 562
10.57. Noetherian graded rings 566
10.58. Noetherian local rings 568
10.59. Dimension 571
10.60. Applications of dimension theory 575
10.61. Support and dimension of modules 576
10.62. Associated primes 577
10.63. Symbolic powers 581
10.64. Relative assassin 581
10.65. Weakly associated primes 584
10.66. Embedded primes 588
10.67. Regular sequences 589
10.68. Quasi-regular sequences 591
10.69. Blow up algebras 594
10.70. Ext groups 596
10.71. Depth 599
10.72. Functorialities for Ext 601
10.73. An application of Ext groups 602
10.74. Tor groups and flatness 603
10.75. Functorialities for Tor 608
10.76. Projective modules 608
10.77. Finite projective modules 610
10.78. Open loci defined by module maps 613
10.79. Faithfully flat descent for projectivity of modules 614
10.80. Characterizing flatness 615
10.81. Universally injective module maps 617
10.82. Descent for finite projective modules 623
10.83. Transfinite d´evissage of modules 624
10.84. Projective modules over a local ring 626
10.85. Mittag-Leffler systems 627
10.86. Inverse systems 629
10.87. Mittag-Leffler modules 629
10.88. Interchanging direct products with tensor 634
10.89. Coherent rings 638
10.90. Examples and non-examples of Mittag-Leffler modules 640
10.91. Countably generated Mittag-Leffler modules 642
10.92. Characterizing projective modules 644
10.93. Ascending properties of modules 645