Table Of ContentStacks Project
Version b0f88a03, compiled on Jan 04, 2020.
Thefollowingpeoplehavecontributedtothiswork: KianAbolfazlian,DanAbramovich,
Juan Pablo Acosta Lopez, Shishir Agrawal, Eric Ahlqvist, Jarod Alper, Johannes
Anschuetz,BenjaminAntieau,KoAoki(青木孔),KonstantinArdakov,DimaArinkin,
Aravind Asok, Dario Balboni, Adrian Barquero-Sanchez, Owen Barrett, Giulia
Battiston, Hanno Becker, Mark Behrens, Pieter Belmans, Olivier Benoist, Daniel
Bergh, Bhargav Bhatt, Wessel Bindt, Chris Birkbeck, Ingo Blechschmidt, Mark
Bowron, Lucas Braune, Thomas Brazelton, Martin Bright, David Brown, Kong
Bochao, Sebastian Bozlee, Niels Borne, Félix Baril Boudreau, Elyes Boughattas,
Ragnar-OlafBuchweitz,KevinBuzzard,JakubByszewski,ZhaodongCai,Nicholas
Camacho, Samir Canning, Robert Cardona, Nuno Cardoso, Scott Carnahan, Kęs-
tutis Česnavičius, Antoine Chambert-Loir, Will Chen, William Chen, Raymond
Cheng (程毅), Filip Chindea, Chung Ching, Nava Chitrik, Fraser Chiu, Kat Chris-
tianson, Patrick Chu, Benjamin Voulgaris Church, Dustin Clausen, Jérémy Co-
choy, Johan Commelin, Brian Conrad, David Corwin, Sean Cotner, Pavel Čoupek,
Peadar Coyle, Moises Herradon Cueto, Marco D’Addezio, Rankeya Datta, Aise
Johan de Jong, Chiara Damiolini, Matt DeLand, Ashwin Deopurkar, Maarten De-
rickx, Neeraj Deshmukh, Benjamin Diamond, Claus Diem, Ajneet Dhillon, Daniel
Disegni, Joel Dodge, Daniel Dore, Peng Du, Taylor Dupuy, Bas Edixhoven, Jonas
Ehrhard, Alexander Palen Ellis, Matthew Emerton, Aras Ergus, Andrew Fanoe,
Maxim Fedorchuck, Hu Fei, Peter Fleischmann, Dan Fox, Cameron Franc, Dra-
gos Fratila, Robert Friedman, Robert Furber, Ofer Gabber, Juan Sebastian Gai-
tan, Lennart Galinat, Martin Gallauer, Luis Garcia, Xu Gao, Toby Gee, Anton
Geraschenko, Daniel Gerigk, Alberto Gioia, Charles Godfrey, Julia Ramos Gonza-
lez, Jean-Pierre Gourdot, Matt Grimes, Darij Grinberg, Jonathan Gruner, Yuzhou
Gu, Zeshen Gu, Quentin Guignard, Albert Gunawan, Joseph Gunther, Haoyang
Guo, Andrei Halanay, Yatir Halevi, Jack Hall, Daniel Halpern-Leistner, Linus
Hamann,MinsikHan,XueHang,DavidHansen,YunHao,MichaelHarris,William
Hart, Philipp Hartwig, Mohamed Hashi, Olivier Haution, Tongmu He, Hadi He-
dayatzadeh, Florian Heiderich, Daniel Heiss, Jeremiah Heller, Reimundo Heluani,
KristenHendricks,AronHeleodoro,ChristianHildebrandt,FraserHiu,QuocP.Ho,
Manuel Hoff, Amit Hogadi, David Holmes, Andreas Holmstrom, Tim Holzschuh,
Ray Hoobler, John Hosack, Xiaowen Hu, Yuhao Huang, Yu-Liang Huang, Ariyan
Javanpeykar, Lena Min Ji, Peter Johnson, Mattias Jonsson, Grayson Jorgenson,
Christian Kappen, Hayama Kazuma (羽山籍真), Kiran Kedlaya, Timo Keller,
Adeel Ahmad Khan, Keenan Kidwell, Ammar Kilic, Andrew Kiluk, Dongryul
Kim, Lars Kindler, János Kollár, Vladimir Kondratjew, Dmitry Korb, Praphulla
Koushik, Sándor Kovács, Emmanuel Kowalski, Girish Kulkarni, Matthias Kum-
merer, Manoj Kummini, Daniel Krashen, Oleksandr Kravets, Brian Lawrence, Ge-
offrey Lee, Min Lee, Simon Pepin Lehalleur, Tobi Lehman, Florian Lengyel, Pak-
HinLee, BrandonLevin, DanielLevine, PaulLessard, MaoLi, ShizhangLi(李璋),
Wen-Wei Li, Carl Lian, Max Lieblich, Bronson Lim, David Benjamin Lim, Joseph
Lipman, Daniel Litt, Huaxin Liu, Hsing Liu, Linyuan Liu, Qing Liu, Xuande Liu,
Zeyu Liu, David Loeffler, Davide Lombardo, Dino Lorenzini, Weixiao Lu, David
Lubicz, Qixiao Ma, Zachary Maddock, Mohammed Mammeri, Zhouhang Mao,
Sonja Mapes, Florent Martin, Kazuki Masugi (馬杉和貴), Akhil Mathew, Fan-
junMeng,DanielMiller,BenMoonen,YogeshMore,LaurentMoret-Bailly,Maxim
Mornev, Jackson Morrow, Nicolas Müller, Alapan Mukhopadhyay, Takumi Mu-
rayama, Yusuf Mustopa, David Mykytyn, Michael Neururer, Josh Nichols-Barrer,
Kien Nguyen, Thomas Nyberg, Masahiro Ohno, Catherine O’Neil, Noah Olan-
der, MartinOlsson, FabriceOrgogozo, BrianOsserman, MarisOzols, MikePaluch,
Thanos Papaioannou, Roland Paulin, Rakesh Pawar, Dmitrii Pedchenko, Peter
Percival, Alex Perry, Maik Pickl, Dmitrii Pirozhkov, Gregor Pohl, Bjorn Poonen,
AnatolyPreygel,ArtemPrihodko,ThibautPugin,SouparnaPurohit,YouQi,Hao-
nan Qu (曲昊男), Eamon Quinlan, Ryan Reich, Emanuel Reinecke, Charles Rezk,
AliceRizzardo,DamienRobert,DavidRoberts,JobRock,HermanRohrbach,Fred
Rohrer, Matthieu Romagny, Joe Ross, Julius Ross, Apurba Kumar Roy, Rob Roy,
Yairon Cid Ruiz, Nithi Rungtanapirom, David Rydh, Carles Sáez, Jyoti Prakash
Saha,BerenSanders,ThéodeOliveiraSantos,SteffenSagave,WillSawin,Federico
Scavia, Alexander Schmidt, Olaf Schnürer, Jakob Scholbach, Rene Schoof, Jaakko
Seppala, Michele Serra, Emre Sertoz, Chung-chieh Shan, Arpith Shanbhag, Liran
Shaul,CheShen,MinseonShin,JeroenSijsling,JohnSmith,ThomasSmith,Dylan
Spence, David Speyer, Tanya Kaushal Srivastava, Axel Stäbler, Jason Starr, Matt
Stevenson, Thierry Stulemeijer, Takashi Suzuki, Lenny Taelman, Mattia Talpo,
AbolfazlTarizadeh,JohnTate,DavidTaylor,TitusTeodorescu,MichaelThaddeus,
Stulemeijer Thierry, Shabalin Timofey, Valery Tolstov, Alex Torzewski, Dajano
Tossici, Burt Totaro, Minh-Tien Tran, David Tweedle, Ravi Vakil, Michel Van den
Bergh,TheovandenBogaart,MatthévanderLee,JeroenvanderMeer,Remyvan
Dobben de Bruyn, Kevin Ventullo, Hendrik Verhoek, Antoine Vezier, Erik Visse,
Angelo Vistoli, Konrad Voelkel, Fred Vu, Rishi Vyas, James Waldron, Hua Wang,
Jonathan Wang, Matthew Ward, Evan Warner, Nils Waßmuth, John Watterlond,
Torsten Wedhorn, Dario Weissmann, Ian Whitehead, Jonathan Wise, Junho Won,
WilliamWright,DominicWynter,DaxinXu,FeiXu,JiachangXu,JunyanXu,Wei
Xu, Qijun Yan, Mengxue Yang, Amnon Yekutieli, Alex Youcis, Jize Yu, John Yu,
Koito Yuu, Felipe Zaldivar, Bogdan Zavyalov, Maciek Zdanowicz, Dingxin Zhang,
Keke Zhang, Lei Zhang, Robin Zhang, Zhe Zhang, Zhiyu Zhang, Jiayu Zhao, Yifei
Zhao,YuZhao,FanZheng,WeizheZheng,AnfangZhou,YichengZhou,FanZhou,
Wouter Zomervrucht, Runpu Zong, Jeroen Zuiddam, David Zureick-Brown.
3
Copyright (C) 2005 -- 2018 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".
Part 1
Preliminaries
CHAPTER 1
Introduction
0000 1.1. Overview
0001 Besides the book by Laumon and Moret-Bailly, see [LMB00], and the work (in
progress) by Fulton et al, we think there is a place for an open source textbook
on algebraic stacks and the algebraic geometry that is needed to define them. The
Stacks Project attempts to do this by building the foundations starting with com-
mutative algebra and proceeding via the theory of schemes and algebraic spaces to
a comprehensive foundation for the theory of algebraic stacks.
Weexpectthismaterialtobereadonlineasakeyfeaturearethehyperlinksgiving
quick access to internal references spread over many different pages. If you use an
embedded pdf or dvi viewer in your browser, the cross file links should work.
Thisprojectisacollaborativeeffortandweencourageyoutohelpout. Pleaseemail
any typos or errors you find while reading or any suggestions, additional material,
or examples you have to [email protected]. You can download a tarball
containing all source files, extract, run make, and use a dvi or pdf viewer locally.
Please feel free to edit the LaTeX files and email your improvements.
1.2. Attribution
06LB The scope of this work is such that it is a daunting task to attribute correctly and
succinctlyallofthosemathematicianswhoseworkhasledtothedevelopmentofthe
theory we try to explain here. We hope eventually to generate enough community
interest to find contributors willing to write sections with historical remarks for
each and every chapter.
Thosewhocontributedtothisworkarelistedonthetitlepageofthebookversionof
thisworkandonline. Herewewouldliketonameaselectionofmajorcontributions:
(1) Jarod Alper contributed a chapter discussing the literature on algebraic
stacks, see Guide to Literature, Section 109.1.
(2) Bhargav Bhatt wrote the initial version of a chapter on étale morphisms
of schemes, see Étale Morphisms, Section 41.1.
(3) BhargavBhattwrotetheinitialversionofMoreonAlgebra,Section15.82.
(4) Kiran Kedlaya contributed the initial writeup of Descent, Section 35.4.
(5) The initial versions of
(a) Algebra, Section 10.27,
(b) Injectives, Section 19.2, and
(c) the chapter on fields, see Fields, Section 9.1.
are from The CRing Project, courtesy of Akhil Mathew et al.
(6) AlexPerrywrotethematerialonprojectivemodules,Mittag-Lefflermod-
ules, including the proof of Algebra, Theorem 10.94.5.
5
1.3. OTHER CHAPTERS 6
(7) AlexPerrywrotethechapterondeformationtheoryalaSchlessingerand
Rim, see Formal Deformation Theory, Section 87.1.
(8) Thibaut Pugin, Zachary Maddock and Min Lee took notes for a course
which formed the basis for a chapter on étale cohomology and a chapter
on the trace formula. See Étale Cohomology, Section 57.1 and The Trace
Formula, Section 61.1.
(9) David Rydh has contributed many helpful comments, pointed out several
mistakes, helped out in an essential way with the material on residual
gerbes, and was the originator for the material in More on Groupoids in
Spaces, Sections 76.12 and 76.15.
(10) Burt Totaro contributed Examples, Sections 107.57, 107.58, and Proper-
ties of Stacks, Section 97.12.
(11) The chapter on pro-étale cohomology, see Pro-étale Cohomology, Section
59.1, is taken from a paper by Bhargav Bhatt and Peter Scholze.
(12) Bhargav Bhatt contributed Examples, Sections 107.64 and 107.68.
(13) Ofer Gabber found mistakes, contributed corrections and he contributed
Varieties, Lemma 33.7.17, Formal Spaces, Lemma 84.9.5, the material in
MoreonGroupoids,Section40.15,themainresultofPropertiesofSpaces,
Section 63.17, and the proof of More on Flatness, Proposition 38.25.13.
(14) János Kollár contributed Algebra, Lemma 10.118.2 and Local Cohomol-
ogy, Proposition 51.8.7.
(15) KiranKedlayawrotetheinitialversionofMoreonAlgebra,Section15.83.
(16) Matthew Emerton, Toby Gee, and Brandon Levin contributed some re-
sultsonthickenings, inparticularMoreonMorphismsofStacks, Lemmas
103.3.7, 103.3.8, and 103.3.9.
(17) Lena Min Ji wrote the initial version of More on Algebra, Section 15.113.
(18) Matthew Emerton and Toby Gee wrote the initial versions of Geometry
of Stacks, Sections 104.3 and 104.5.
1.3. Other chapters
Preliminaries (17) Sheaves of Modules
(18) Modules on Sites
(1) Introduction (19) Injectives
(2) Conventions (20) Cohomology of Sheaves
(3) Set Theory (21) Cohomology on Sites
(4) Categories (22) Differential Graded Algebra
(5) Topology (23) Divided Power Algebra
(6) Sheaves on Spaces (24) Differential Graded Sheaves
(7) Sites and Sheaves (25) Hypercoverings
(8) Stacks
Schemes
(9) Fields
(10) Commutative Algebra (26) Schemes
(11) Brauer Groups (27) Constructions of Schemes
(12) Homological Algebra (28) Properties of Schemes
(13) Derived Categories (29) Morphisms of Schemes
(14) Simplicial Methods (30) Cohomology of Schemes
(15) More on Algebra (31) Divisors
(16) Smoothing Ring Maps (32) Limits of Schemes
1.3. OTHER CHAPTERS 7
(33) Varieties (76) More on Groupoids in Spaces
(34) Topologies on Schemes (77) Bootstrap
(35) Descent (78) Pushouts of Algebraic Spaces
(36) Derived Categories of Schemes Topics in Geometry
(37) More on Morphisms
(79) Chow Groups of Spaces
(38) More on Flatness
(80) Quotients of Groupoids
(39) Groupoid Schemes
(81) MoreonCohomologyofSpaces
(40) More on Groupoid Schemes
(82) Simplicial Spaces
(41) Étale Morphisms of Schemes
(83) Duality for Spaces
Topics in Scheme Theory (84) Formal Algebraic Spaces
(42) Chow Homology (85) Restricted Power Series
(43) Intersection Theory (86) Resolution of Surfaces Revis-
(44) Picard Schemes of Curves ited
(45) Weil Cohomology Theories Deformation Theory
(46) Adequate Modules
(87) Formal Deformation Theory
(47) Dualizing Complexes
(88) Deformation Theory
(48) Duality for Schemes
(89) The Cotangent Complex
(49) Discriminants and Differents
(90) Deformation Problems
(50) de Rham Cohomology
Algebraic Stacks
(51) Local Cohomology
(91) Algebraic Stacks
(52) Algebraic and Formal Geome-
(92) Examples of Stacks
try
(93) Sheaves on Algebraic Stacks
(53) Algebraic Curves
(94) Criteria for Representability
(54) Resolution of Surfaces
(95) Artin’s Axioms
(55) Semistable Reduction
(96) Quot and Hilbert Spaces
(56) Fundamental Groups of
(97) Properties of Algebraic Stacks
Schemes
(98) MorphismsofAlgebraicStacks
(57) Étale Cohomology
(99) Limits of Algebraic Stacks
(58) Crystalline Cohomology
(100) Cohomology of Algebraic
(59) Pro-étale Cohomology
Stacks
(60) More Étale Cohomology
(101) Derived Categories of Stacks
(61) The Trace Formula
(102) Introducing Algebraic Stacks
Algebraic Spaces
(103) More on Morphisms of Stacks
(62) Algebraic Spaces
(104) The Geometry of Stacks
(63) Properties of Algebraic Spaces
Topics in Moduli Theory
(64) MorphismsofAlgebraicSpaces
(105) Moduli Stacks
(65) Decent Algebraic Spaces
(106) Moduli of Curves
(66) Cohomology of Algebraic
Spaces Miscellany
(67) Limits of Algebraic Spaces (107) Examples
(68) Divisors on Algebraic Spaces (108) Exercises
(69) Algebraic Spaces over Fields (109) Guide to Literature
(70) TopologiesonAlgebraicSpaces (110) Desirables
(71) Descent and Algebraic Spaces (111) Coding Style
(72) Derived Categories of Spaces (112) Obsolete
(73) More on Morphisms of Spaces (113) GNU Free Documentation Li-
(74) Flatness on Algebraic Spaces cense
(75) Groupoids in Algebraic Spaces (114) Auto Generated Index
CHAPTER 2
Conventions
0002 2.1. Comments
0003 Thephilosophybehindtheconventionsusedinwritingthesedocumentsistochoose
those conventions that work.
2.2. Set theory
0004 We use Zermelo-Fraenkel set theory with the axiom of choice. See [Kun83]. We
do not use universes (different from SGA4). We do not stress set-theoretic issues,
but we make sure everything is correct (of course) and so we do not ignore them
either.
2.3. Categories
0005 A category C consists of a set of objects and, for each pair of objects, a set of
morphismsbetweenthem. Inotherwords, itiswhatiscalleda“small”categoryin
other texts. We will use “big” categories (categories whose objects form a proper
class) as well, but only those that are listed in Categories, Remark 4.2.2.
2.4. Algebra
0006 In these notes a ring is a commutative ring with a 1. Hence the category of rings
has an initial object Z and a final object {0} (this is the unique ring where 1=0).
Modules are assumed unitary. See [Eis95].
2.5. Notation
055X The natural integers are elements of N = {1,2,3,...}. The integers are elements
of Z = {...,−2,−1,0,1,2,...}. The field of rational numbers is denoted Q. The
field of real numbers is denoted R. The field of complex numbers is denoted C.
2.6. Other chapters
Preliminaries (10) Commutative Algebra
(11) Brauer Groups
(1) Introduction
(12) Homological Algebra
(2) Conventions
(13) Derived Categories
(3) Set Theory
(14) Simplicial Methods
(4) Categories
(15) More on Algebra
(5) Topology
(16) Smoothing Ring Maps
(6) Sheaves on Spaces
(17) Sheaves of Modules
(7) Sites and Sheaves
(18) Modules on Sites
(8) Stacks
(19) Injectives
(9) Fields
8
2.6. OTHER CHAPTERS 9
(20) Cohomology of Sheaves (62) Algebraic Spaces
(21) Cohomology on Sites (63) Properties of Algebraic Spaces
(22) Differential Graded Algebra (64) MorphismsofAlgebraicSpaces
(23) Divided Power Algebra (65) Decent Algebraic Spaces
(24) Differential Graded Sheaves (66) Cohomology of Algebraic
(25) Hypercoverings Spaces
(67) Limits of Algebraic Spaces
Schemes
(68) Divisors on Algebraic Spaces
(26) Schemes
(69) Algebraic Spaces over Fields
(27) Constructions of Schemes
(70) TopologiesonAlgebraicSpaces
(28) Properties of Schemes
(71) Descent and Algebraic Spaces
(29) Morphisms of Schemes
(72) Derived Categories of Spaces
(30) Cohomology of Schemes
(73) More on Morphisms of Spaces
(31) Divisors
(74) Flatness on Algebraic Spaces
(32) Limits of Schemes
(75) Groupoids in Algebraic Spaces
(33) Varieties
(76) More on Groupoids in Spaces
(34) Topologies on Schemes
(77) Bootstrap
(35) Descent
(78) Pushouts of Algebraic Spaces
(36) Derived Categories of Schemes
Topics in Geometry
(37) More on Morphisms
(38) More on Flatness (79) Chow Groups of Spaces
(39) Groupoid Schemes (80) Quotients of Groupoids
(40) More on Groupoid Schemes (81) MoreonCohomologyofSpaces
(41) Étale Morphisms of Schemes (82) Simplicial Spaces
(83) Duality for Spaces
Topics in Scheme Theory
(84) Formal Algebraic Spaces
(42) Chow Homology (85) Restricted Power Series
(43) Intersection Theory (86) Resolution of Surfaces Revis-
(44) Picard Schemes of Curves ited
(45) Weil Cohomology Theories
Deformation Theory
(46) Adequate Modules
(47) Dualizing Complexes (87) Formal Deformation Theory
(48) Duality for Schemes (88) Deformation Theory
(49) Discriminants and Differents (89) The Cotangent Complex
(50) de Rham Cohomology (90) Deformation Problems
(51) Local Cohomology Algebraic Stacks
(52) Algebraic and Formal Geome-
(91) Algebraic Stacks
try
(92) Examples of Stacks
(53) Algebraic Curves
(93) Sheaves on Algebraic Stacks
(54) Resolution of Surfaces
(94) Criteria for Representability
(55) Semistable Reduction
(95) Artin’s Axioms
(56) Fundamental Groups of
(96) Quot and Hilbert Spaces
Schemes
(97) Properties of Algebraic Stacks
(57) Étale Cohomology
(98) MorphismsofAlgebraicStacks
(58) Crystalline Cohomology
(99) Limits of Algebraic Stacks
(59) Pro-étale Cohomology
(100) Cohomology of Algebraic
(60) More Étale Cohomology
Stacks
(61) The Trace Formula
(101) Derived Categories of Stacks
Algebraic Spaces (102) Introducing Algebraic Stacks
2.6. OTHER CHAPTERS 10
(103) More on Morphisms of Stacks (108) Exercises
(104) The Geometry of Stacks (109) Guide to Literature
(110) Desirables
Topics in Moduli Theory
(111) Coding Style
(105) Moduli Stacks
(112) Obsolete
(106) Moduli of Curves
(113) GNU Free Documentation Li-
Miscellany cense
(107) Examples (114) Auto Generated Index
