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

5338 Pages·2016·25 MB·English
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Preview The Stacks Project: Algebraic Stacks and the Algebraic Geometry Needed to Define Them

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

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This is the home page of the Stacks project. It is an open source textbook and reference work on algebraic stacks and the algebraic geometry needed to define them. The Stacks project started in 2005. The initial idea was for it to be a collaborative web-based project with the aim of writing an intro
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