Stacks Project
Free

Stacks Project

By Stacks Project
Free
Book Description

The Stacks project is an open source textbook and reference work on algebraic stacks and the algebraic geometry needed to define them. For more general information and the latest revision, see the extensive about page.



Archived Version is # bd7e5af, compiled on Sep 07, 2015.

Table of Contents
  • Chapter 1. Introduction
    • 1.1. Overview
    • 1.2. Attribution
    • 1.3. Other chapters
  • Chapter 2. Conventions
    • 2.1. Comments
    • 2.2. Set theory
    • 2.3. Categories
    • 2.4. Algebra
    • 2.5. Notation
    • 2.6. Other chapters
  • Chapter 3. Set Theory
    • 3.1. Introduction
    • 3.2. Everything is a set
    • 3.3. Classes
    • 3.4. Ordinals
    • 3.5. The hierarchy of sets
    • 3.6. Cardinality
    • 3.7. Cofinality
    • 3.8. Reflection principle
    • 3.9. Constructing categories of schemes
    • 3.10. Sets with group action
    • 3.11. Coverings of a site
    • 3.12. Abelian categories and injectives
    • 3.13. Other chapters
  • Chapter 4. Categories
    • 4.1. Introduction
    • 4.2. Definitions
    • 4.3. Opposite Categories and the Yoneda Lemma
    • 4.4. Products of pairs
    • 4.5. Coproducts of pairs
    • 4.6. Fibre products
    • 4.7. Examples of fibre products
    • 4.8. Fibre products and representability
    • 4.9. Pushouts
    • 4.10. Equalizers
    • 4.11. Coequalizers
    • 4.12. Initial and final objects
    • 4.13. Monomorphisms and Epimorphisms
    • 4.14. Limits and colimits
    • 4.15. Limits and colimits in the category of sets
    • 4.16. Connected limits
    • 4.17. Cofinal and initial categories
    • 4.18. Finite limits and colimits
    • 4.19. Filtered colimits
    • 4.20. Cofiltered limits
    • 4.21. Limits and colimits over partially ordered sets
    • 4.22. Essentially constant systems
    • 4.23. Exact functors
    • 4.24. Adjoint functors
    • 4.25. A criterion for representability
    • 4.26. Localization in categories
    • 4.27. Formal properties
    • 4.28. 2-categories
    • 4.29. (2, 1)-categories
    • 4.30. 2-fibre products
    • 4.31. Categories over categories
    • 4.32. Fibred categories
    • 4.33. Inertia
    • 4.34. Categories fibred in groupoids
    • 4.35. Presheaves of categories
    • 4.36. Presheaves of groupoids
    • 4.37. Categories fibred in sets
    • 4.38. Categories fibred in setoids
    • 4.39. Representable categories fibred in groupoids
    • 4.40. Representable 1-morphisms
    • 4.41. Other chapters
  • Chapter 5. Topology
    • 5.1. Introduction
    • 5.2. Basic notions
    • 5.3. Hausdorff spaces
    • 5.4. Bases
    • 5.5. Submersive maps
    • 5.6. Connected components
    • 5.7. Irreducible components
    • 5.8. Noetherian topological spaces
    • 5.9. Krull dimension
    • 5.10. Codimension and catenary spaces
    • 5.11. Quasi-compact spaces and maps
    • 5.12. Locally quasi-compact spaces
    • 5.13. Limits of spaces
    • 5.14. Constructible sets
    • 5.15. Constructible sets and Noetherian spaces
    • 5.16. Characterizing proper maps
    • 5.17. Jacobson spaces
    • 5.18. Specialization
    • 5.19. Dimension functions
    • 5.20. Nowhere dense sets
    • 5.21. Profinite spaces
    • 5.22. Spectral spaces
    • 5.23. Limits of spectral spaces
    • 5.24. Stone-Cech compactification
    • 5.25. Extremally disconnected spaces
    • 5.26. Miscellany
    • 5.27. Partitions and stratifications
    • 5.28. Colimits of spaces
    • 5.29. Topological groups, rings, modules
    • 5.30. Other chapters
  • Chapter 6. Sheaves on Spaces
    • 6.1. Introduction
    • 6.2. Basic notions
    • 6.3. Presheaves
    • 6.4. Abelian presheaves
    • 6.5. Presheaves of algebraic structures
    • 6.6. Presheaves of modules
    • 6.7. Sheaves
    • 6.8. Abelian sheaves
    • 6.9. Sheaves of algebraic structures
    • 6.10. Sheaves of modules
    • 6.11. Stalks
    • 6.12. Stalks of abelian presheaves
    • 6.13. Stalks of presheaves of algebraic structures
    • 6.14. Stalks of presheaves of modules
    • 6.15. Algebraic structures
    • 6.16. Exactness and points
    • 6.17. Sheafification
    • 6.18. Sheafification of abelian presheaves
    • 6.19. Sheafification of presheaves of algebraic structures
    • 6.20. Sheafification of presheaves of modules
    • 6.21. Continuous maps and sheaves
    • 6.22. Continuous maps and abelian sheaves
    • 6.23. Continuous maps and sheaves of algebraic structures
    • 6.24. Continuous maps and sheaves of modules
    • 6.25. Ringed spaces
    • 6.26. Morphisms of ringed spaces and modules
    • 6.27. Skyscraper sheaves and stalks
    • 6.28. Limits and colimits of presheaves
    • 6.29. Limits and colimits of sheaves
    • 6.30. Bases and sheaves
    • 6.31. Open immersions and (pre)sheaves
    • 6.32. Closed immersions and (pre)sheaves
    • 6.33. Glueing sheaves
    • 6.34. Other chapters
  • Chapter 7. Sites and Sheaves
    • 7.1. Introduction
    • 7.2. Presheaves
    • 7.3. Injective and surjective maps of presheaves
    • 7.4. Limits and colimits of presheaves
    • 7.5. Functoriality of categories of presheaves
    • 7.6. Sites
    • 7.7. Sheaves
    • 7.8. Families of morphisms with fixed target
    • 7.9. The example of G-sets
    • 7.10. Sheafification
    • 7.11. Quasi-compact objects and colimits
    • 7.12. Injective and surjective maps of sheaves
    • 7.13. Representable sheaves
    • 7.14. Continuous functors
    • 7.15. Morphisms of sites
    • 7.16. Topoi
    • 7.17. G-sets and morphisms
    • 7.18. More functoriality of presheaves
    • 7.19. Cocontinuous functors
    • 7.20. Cocontinuous functors and morphisms of topoi
    • 7.21. Cocontinuous functors which have a right adjoint
    • 7.22. Cocontinuous functors which have a left adjoint
    • 7.23. Existence of lower shriek
    • 7.24. Localization
    • 7.25. Glueing sheaves
    • 7.26. More localization
    • 7.27. Localization and morphisms
    • 7.28. Morphisms of topoi
    • 7.29. Localization of topoi
    • 7.30. Localization and morphisms of topoi
    • 7.31. Points
    • 7.32. Constructing points
    • 7.33. Points and morphisms of topoi
    • 7.34. Localization and points
    • 7.35. 2-morphisms of topoi
    • 7.36. Morphisms between points
    • 7.37. Sites with enough points
    • 7.38. Criterion for existence of points
    • 7.39. Weakly contractible objects
    • 7.40. Exactness properties of pushforward
    • 7.41. Almost cocontinuous functors
    • 7.42. Subtopoi
    • 7.43. Sheaves of algebraic structures
    • 7.44. Pullback maps
    • 7.45. Topologies
    • 7.46. The topology defined by a site
    • 7.47. Sheafification in a topology
    • 7.48. Topologies and sheaves
    • 7.49. Topologies and continuous functors
    • 7.50. Points and topologies
    • 7.51. Other chapters
  • Chapter 8. Stacks
    • 8.1. Introduction
    • 8.2. Presheaves of morphisms associated to fibred categories
    • 8.3. Descent data in fibred categories
    • 8.4. Stacks
    • 8.5. Stacks in groupoids
    • 8.6. Stacks in setoids
    • 8.7. The inertia stack
    • 8.8. Stackification of fibred categories
    • 8.9. Stackification of categories fibred in groupoids
    • 8.10. Inherited topologies
    • 8.11. Gerbes
    • 8.12. Functoriality for stacks
    • 8.13. Stacks and localization
    • 8.14. Other chapters
  • Chapter 9. Fields
    • 9.1. Introduction
    • 9.2. Basic definitions
    • 9.3. Examples of fields
    • 9.4. Vector spaces
    • 9.5. The characteristic of a field
    • 9.6. Field extensions
    • 9.7. Finite extensions
    • 9.8. Algebraic extensions
    • 9.9. Minimal polynomials
    • 9.10. Algebraic closure
    • 9.11. Relatively prime polynomials
    • 9.12. Separable extensions
    • 9.13. Purely inseparable extensions
    • 9.14. Normal extensions
    • 9.15. Splitting fields
    • 9.16. Roots of unity
    • 9.17. Finite fields
    • 9.18. Primitive elements
    • 9.19. Trace and norm
    • 9.20. Galois theory
    • 9.21. Infinite Galois theory
    • 9.22. The complex numbers
    • 9.23. Kummer extensions
    • 9.24. Artin-Schreier extensions
    • 9.25. Transcendence
    • 9.26. Linearly disjoint extensions
    • 9.27. Review
    • 9.28. Other chapters
  • Chapter 10. Commutative Algebra
    • 10.1. Introduction
    • 10.2. Conventions
    • 10.3. Basic notions
    • 10.4. Snake lemma
    • 10.5. Finite modules and finitely presented modules
    • 10.6. Ring maps of finite type and of finite presentation
    • 10.7. Finite ring maps
    • 10.8. Colimits
    • 10.9. Localization
    • 10.10. Internal Hom
    • 10.11. Tensor products
    • 10.12. Tensor algebra
    • 10.13. Base change
    • 10.14. Miscellany
    • 10.15. Cayley-Hamilton
    • 10.16. The spectrum of a ring
    • 10.17. Local rings
    • 10.18. The Jacobson radical of a ring
    • 10.19. Nakayama's lemma
    • 10.20. Open and closed subsets of spectra
    • 10.21. Connected components of spectra
    • 10.22. Glueing functions
    • 10.23. More glueing results
    • 10.24. Zerodivisors and total rings of fractions
    • 10.25. Irreducible components of spectra
    • 10.26. Examples of spectra of rings
    • 10.27. A meta-observation about prime ideals
    • 10.28. Images of ring maps of finite presentation
    • 10.29. More on images
    • 10.30. Noetherian rings
    • 10.31. Locally nilpotent ideals
    • 10.32. Curiosity
    • 10.33. Hilbert Nullstellensatz
    • 10.34. Jacobson rings
    • 10.35. Finite and integral ring extensions
    • 10.36. Normal rings
    • 10.37. Going down for integral over normal
    • 10.38. Flat modules and flat ring maps
    • 10.39. Supports and annihilators
    • 10.40. Going up and going down
    • 10.41. Separable extensions
    • 10.42. Geometrically reduced algebras
    • 10.43. Separable extensions, continued
    • 10.44. Perfect fields
    • 10.45. Universal homeomorphisms
    • 10.46. Geometrically irreducible algebras
    • 10.47. Geometrically connected algebras
    • 10.48. Geometrically integral algebras
    • 10.49. Valuation rings
    • 10.50. More Noetherian rings
    • 10.51. Length
    • 10.52. Artinian rings
    • 10.53. Homomorphisms essentially of finite type
    • 10.54. K-groups
    • 10.55. Graded rings
    • 10.56. Proj of a graded ring
    • 10.57. Noetherian graded rings
    • 10.58. Noetherian local rings
    • 10.59. Dimension
    • 10.60. Applications of dimension theory
    • 10.61. Support and dimension of modules
    • 10.62. Associated primes
    • 10.63. Symbolic powers
    • 10.64. Relative assassin
    • 10.65. Weakly associated primes
    • 10.66. Embedded primes
    • 10.67. Regular sequences
    • 10.68. Quasi-regular sequences
    • 10.69. Blow up algebras
    • 10.70. Ext groups
    • 10.71. Depth
    • 10.72. Functorialities for Ext
    • 10.73. An application of Ext groups
    • 10.74. Tor groups and flatness
    • 10.75. Functorialities for Tor
    • 10.76. Projective modules
    • 10.77. Finite projective modules
    • 10.78. Open loci defined by module maps
    • 10.79. Faithfully flat descent for projectivity of modules
    • 10.80. Characterizing flatness
    • 10.81. Universally injective module maps
    • 10.82. Descent for finite projective modules
    • 10.83. Transfinite dévissage of modules
    • 10.84. Projective modules over a local ring
    • 10.85. Mittag-Leffler systems
    • 10.86. Inverse systems
    • 10.87. Mittag-Leffler modules
    • 10.88. Interchanging direct products with tensor
    • 10.89. Coherent rings
    • 10.90. Examples and non-examples of Mittag-Leffler modules
    • 10.91. Countably generated Mittag-Leffler modules
    • 10.92. Characterizing projective modules
    • 10.93. Ascending properties of modules
    • 10.94. Descending properties of modules
    • 10.95. Completion
    • 10.96. Completion for Noetherian rings
    • 10.97. Taking limits of modules
    • 10.98. Criteria for flatness
    • 10.99. Base change and flatness
    • 10.100. Flatness criteria over Artinian rings
    • 10.101. What makes a complex exact?
    • 10.102. Cohen-Macaulay modules
    • 10.103. Cohen-Macaulay rings
    • 10.104. Catenary rings
    • 10.105. Regular local rings
    • 10.106. Epimorphisms of rings
    • 10.107. Pure ideals
    • 10.108. Rings of finite global dimension
    • 10.109. Regular rings and global dimension
    • 10.110. Auslander-Buchsbaum
    • 10.111. Homomorphisms and dimension
    • 10.112. The dimension formula
    • 10.113. Dimension of finite type algebras over fields
    • 10.114. Noether normalization
    • 10.115. Dimension of finite type algebras over fields, reprise
    • 10.116. Dimension of graded algebras over a field
    • 10.117. Generic flatness
    • 10.118. Around Krull-Akizuki
    • 10.119. Factorization
    • 10.120. Orders of vanishing
    • 10.121. Quasi-finite maps
    • 10.122. Zariski's Main Theorem
    • 10.123. Applications of Zariski's Main Theorem
    • 10.124. Dimension of fibres
    • 10.125. Algebras and modules of finite presentation
    • 10.126. Colimits and maps of finite presentation
    • 10.127. More flatness criteria
    • 10.128. Openness of the flat locus
    • 10.129. Openness of Cohen-Macaulay loci
    • 10.130. Differentials
    • 10.131. Finite order differential operators
    • 10.132. The naive cotangent complex
    • 10.133. Local complete intersections
    • 10.134. Syntomic morphisms
    • 10.135. Smooth ring maps
    • 10.136. Formally smooth maps
    • 10.137. Smoothness and differentials
    • 10.138. Smooth algebras over fields
    • 10.139. Smooth ring maps in the Noetherian case
    • 10.140. Overview of results on smooth ring maps
    • 10.141. Étale ring maps
    • 10.142. Local homomorphisms
    • 10.143. Integral closure and smooth base change
    • 10.144. Formally unramified maps
    • 10.145. Conormal modules and universal thickenings
    • 10.146. Formally étale maps
    • 10.147. Unramified ring maps
    • 10.148. Henselian local rings
    • 10.149. Serre's criterion for normality
    • 10.150. Formal smoothness of fields
    • 10.151. Constructing flat ring maps
    • 10.152. The Cohen structure theorem
    • 10.153. Japanese rings
    • 10.154. Nagata rings
    • 10.155. Ascending properties
    • 10.156. Descending properties
    • 10.157. Geometrically normal algebras
    • 10.158. Geometrically regular algebras
    • 10.159. Geometrically Cohen-Macaulay algebras
    • 10.160. Colimits and maps of finite presentation, II
    • 10.161. Other chapters
  • Chapter 11. Brauer groups
    • 11.1. Introduction
    • 11.2. Noncommutative algebras
    • 11.3. Wedderburn's theorem
    • 11.4. Lemmas on algebras
    • 11.5. The Brauer group of a field
    • 11.6. Skolem-Noether
    • 11.7. The centralizer theorem
    • 11.8. Splitting fields
    • 11.9. Other chapters
  • Chapter 12. Homological Algebra
    • 12.1. Introduction
    • 12.2. Basic notions
    • 12.3. Preadditive and additive categories
    • 12.4. Karoubian categories
    • 12.5. Abelian categories
    • 12.6. Extensions
    • 12.7. Additive functors
    • 12.8. Localization
    • 12.9. Serre subcategories
    • 12.10. K-groups
    • 12.11. Cohomological delta-functors
    • 12.12. Complexes
    • 12.13. Truncation of complexes
    • 12.14. Homotopy and the shift functor
    • 12.15. Graded objects
    • 12.16. Filtrations
    • 12.17. Spectral sequences
    • 12.18. Spectral sequences: exact couples
    • 12.19. Spectral sequences: differential objects
    • 12.20. Spectral sequences: filtered differential objects
    • 12.21. Spectral sequences: filtered complexes
    • 12.22. Spectral sequences: double complexes
    • 12.23. Injectives
    • 12.24. Projectives
    • 12.25. Injectives and adjoint functors
    • 12.26. Essentially constant systems
    • 12.27. Inverse systems
    • 12.28. Exactness of products
    • 12.29. Other chapters
  • Chapter 13. Derived Categories
    • 13.1. Introduction
    • 13.2. Triangulated categories
    • 13.3. The definition of a triangulated category
    • 13.4. Elementary results on triangulated categories
    • 13.5. Localization of triangulated categories
    • 13.6. Quotients of triangulated categories
    • 13.7. Adjoints for exact functors
    • 13.8. The homotopy category
    • 13.9. Cones and termwise split sequences
    • 13.10. Distinguished triangles in the homotopy category
    • 13.11. Derived categories
    • 13.12. The canonical delta-functor
    • 13.13. Triangulated subcategories of the derived category
    • 13.14. Filtered derived categories
    • 13.15. Derived functors in general
    • 13.16. Derived functors on derived categories
    • 13.17. Higher derived functors
    • 13.18. Injective resolutions
    • 13.19. Projective resolutions
    • 13.20. Right derived functors and injective resolutions
    • 13.21. Cartan-Eilenberg resolutions
    • 13.22. Composition of right derived functors
    • 13.23. Resolution functors
    • 13.24. Functorial injective embeddings and resolution functors
    • 13.25. Right derived functors via resolution functors
    • 13.26. Filtered derived category and injective resolutions
    • 13.27. Ext groups
    • 13.28. Unbounded complexes
    • 13.29. K-injective complexes
    • 13.30. Bounded cohomological dimension
    • 13.31. Derived colimits
    • 13.32. Derived limits
    • 13.33. Generators of triangulated categories
    • 13.34. Compact objects
    • 13.35. Brown representability
    • 13.36. Other chapters
  • Chapter 14. Simplicial Methods
    • 14.1. Introduction
    • 14.2. The category of finite ordered sets
    • 14.3. Simplicial objects
    • 14.4. Simplicial objects as presheaves
    • 14.5. Cosimplicial objects
    • 14.6. Products of simplicial objects
    • 14.7. Fibre products of simplicial objects
    • 14.8. Pushouts of simplicial objects
    • 14.9. Products of cosimplicial objects
    • 14.10. Fibre products of cosimplicial objects
    • 14.11. Simplicial sets
    • 14.12. Truncated simplicial objects and skeleton functors
    • 14.13. Products with simplicial sets
    • 14.14. Hom from simplicial sets into cosimplicial objects
    • 14.15. Hom from cosimplicial sets into simplicial objects
    • 14.16. Internal Hom
    • 14.17. Hom from simplicial sets into simplicial objects
    • 14.18. Splitting simplicial objects
    • 14.19. Coskeleton functors
    • 14.20. Augmentations
    • 14.21. Left adjoints to the skeleton functors
    • 14.22. Simplicial objects in abelian categories
    • 14.23. Simplicial objects and chain complexes
    • 14.24. Dold-Kan
    • 14.25. Dold-Kan for cosimplicial objects
    • 14.26. Homotopies
    • 14.27. Homotopies in abelian categories
    • 14.28. Homotopies and cosimplicial objects
    • 14.29. More homotopies in abelian categories
    • 14.30. Trivial Kan fibrations
    • 14.31. Kan fibrations
    • 14.32. A homotopy equivalence
    • 14.33. Standard resolutions
    • 14.34. Other chapters
  • Chapter 15. More on Algebra
    • 15.1. Introduction
    • 15.2. Advice for the reader
    • 15.3. Stably free modules
    • 15.4. A comment on the Artin-Rees property
    • 15.5. Fibre products of rings
    • 15.6. Fitting ideals
    • 15.7. Lifting
    • 15.8. Henselian pairs
    • 15.9. Auto-associated rings
    • 15.10. Flattening stratification
    • 15.11. Flattening over an Artinian ring
    • 15.12. Flattening over a closed subset of the base
    • 15.13. Flattening over a closed subsets of source and base
    • 15.14. Flattening over a Noetherian complete local ring
    • 15.15. Descent flatness along integral maps
    • 15.16. Torsion free modules
    • 15.17. Reflexive modules
    • 15.18. Content ideals
    • 15.19. Flatness and finiteness conditions
    • 15.20. Blowing up and flatness
    • 15.21. Completion and flatness
    • 15.22. The Koszul complex
    • 15.23. Koszul regular sequences
    • 15.24. Regular ideals
    • 15.25. Local complete intersection maps
    • 15.26. Cartier's equality and geometric regularity
    • 15.27. Geometric regularity
    • 15.28. Topological rings and modules
    • 15.29. Formally smooth maps of topological rings
    • 15.30. Some results on power series rings
    • 15.31. Geometric regularity and formal smoothness
    • 15.32. Regular ring maps
    • 15.33. Ascending properties along regular ring maps
    • 15.34. Permanence of properties under completion
    • 15.35. Permanence of properties under étale maps
    • 15.36. Permanence of properties under henselization
    • 15.37. Field extensions, revisited
    • 15.38. The singular locus
    • 15.39. Regularity and derivations
    • 15.40. Formal smoothness and regularity
    • 15.41. G-rings
    • 15.42. Properties of formal fibres
    • 15.43. Excellent rings
    • 15.44. Abelian categories of modules
    • 15.45. Injective abelian groups
    • 15.46. Injective modules
    • 15.47. Derived categories of modules
    • 15.48. Computing Tor
    • 15.49. Derived tensor product
    • 15.50. Derived change of rings
    • 15.51. Tor independence
    • 15.52. Spectral sequences for Tor
    • 15.53. Products and Tor
    • 15.54. Pseudo-coherent modules
    • 15.55. Tor dimension
    • 15.56. Spectral sequences for Ext
    • 15.57. Projective dimension
    • 15.58. Injective dimension
    • 15.59. Hom complexes
    • 15.60. Derived hom
    • 15.61. Perfect complexes
    • 15.62. Lifting complexes
    • 15.63. Splitting complexes
    • 15.64. Characterizing perfect complexes
    • 15.65. Relatively finitely presented modules
    • 15.66. Relatively pseudo-coherent modules
    • 15.67. Pseudo-coherent and perfect ring maps
    • 15.68. Rlim of abelian groups and modules
    • 15.69. Torsion modules
    • 15.70. Formal glueing of module categories
    • 15.71. The Beauville-Laszlo theorem
    • 15.72. Derived Completion
    • 15.73. Derived completion for a principal ideal
    • 15.74. Derived completion for Noetherian rings
    • 15.75. Taking limits of complexes
    • 15.76. Some evaluation maps
    • 15.77. Miscellany
    • 15.78. Weakly étale ring maps
    • 15.79. Local irreducibility
    • 15.80. Group actions and integral closure
    • 15.81. Ramification theory
    • 15.82. Eliminating ramification
    • 15.83. Picard groups of rings
    • 15.84. Extensions of valuation rings
    • 15.85. Structure of modules over a PID
    • 15.86. Other chapters
  • Chapter 16. Smoothing Ring Maps
    • 16.1. Introduction
    • 16.2. Colimits
    • 16.3. Singular ideals
    • 16.4. Presentations of algebras
    • 16.5. Intermezzo: Néron desingularization
    • 16.6. The lifting problem
    • 16.7. The lifting lemma
    • 16.8. The desingularization lemma
    • 16.9. Warmup: reduction to a base field
    • 16.10. Local tricks
    • 16.11. Separable residue fields
    • 16.12. Inseparable residue fields
    • 16.13. The main theorem
    • 16.14. The approximation property for G-rings
    • 16.15. Approximation for henselian pairs
    • 16.16. Other chapters
  • Chapter 17. Sheaves of Modules
    • 17.1. Introduction
    • 17.2. Pathology
    • 17.3. The abelian category of sheaves of modules
    • 17.4. Sections of sheaves of modules
    • 17.5. Supports of modules and sections
    • 17.6. Closed immersions and abelian sheaves
    • 17.7. A canonical exact sequence
    • 17.8. Modules locally generated by sections
    • 17.9. Modules of finite type
    • 17.10. Quasi-coherent modules
    • 17.11. Modules of finite presentation
    • 17.12. Coherent modules
    • 17.13. Closed immersions of ringed spaces
    • 17.14. Locally free sheaves
    • 17.15. Tensor product
    • 17.16. Flat modules
    • 17.17. Flat morphisms of ringed spaces
    • 17.18. Symmetric and exterior powers
    • 17.19. Internal Hom
    • 17.20. Koszul complexes
    • 17.21. Invertible modules
    • 17.22. Rank and determinant
    • 17.23. Localizing sheaves of rings
    • 17.24. Modules of differentials
    • 17.25. The naive cotangent complex
    • 17.26. Other chapters
  • Chapter 18. Modules on Sites
    • 18.1. Introduction
    • 18.2. Abelian presheaves
    • 18.3. Abelian sheaves
    • 18.4. Free abelian presheaves
    • 18.5. Free abelian sheaves
    • 18.6. Ringed sites
    • 18.7. Ringed topoi
    • 18.8. 2-morphisms of ringed topoi
    • 18.9. Presheaves of modules
    • 18.10. Sheaves of modules
    • 18.11. Sheafification of presheaves of modules
    • 18.12. Morphisms of topoi and sheaves of modules
    • 18.13. Morphisms of ringed topoi and modules
    • 18.14. The abelian category of sheaves of modules
    • 18.15. Exactness of pushforward
    • 18.16. Exactness of lower shriek
    • 18.17. Global types of modules
    • 18.18. Intrinsic properties of modules
    • 18.19. Localization of ringed sites
    • 18.20. Localization of morphisms of ringed sites
    • 18.21. Localization of ringed topoi
    • 18.22. Localization of morphisms of ringed topoi
    • 18.23. Local types of modules
    • 18.24. Basic results on local types of modules
    • 18.25. Closed immersions of ringed topoi
    • 18.26. Tensor product
    • 18.27. Internal Hom
    • 18.28. Flat modules
    • 18.29. Towards constructible modules
    • 18.30. Flat morphisms
    • 18.31. Invertible modules
    • 18.32. Modules of differentials
    • 18.33. Finite order differential operators
    • 18.34. The naive cotangent complex
    • 18.35. Stalks of modules
    • 18.36. Skyscraper sheaves
    • 18.37. Localization and points
    • 18.38. Pullbacks of flat modules
    • 18.39. Locally ringed topoi
    • 18.40. Lower shriek for modules
    • 18.41. Constant sheaves
    • 18.42. Locally constant sheaves
    • 18.43. Other chapters
  • Chapter 19. Injectives
    • 19.1. Introduction
    • 19.2. Baer's argument for modules
    • 19.3. G-modules
    • 19.4. Abelian sheaves on a space
    • 19.5. Sheaves of modules on a ringed space
    • 19.6. Abelian presheaves on a category
    • 19.7. Abelian Sheaves on a site
    • 19.8. Modules on a ringed site
    • 19.9. Embedding abelian categories
    • 19.10. Grothendieck's AB conditions
    • 19.11. Injectives in Grothendieck categories
    • 19.12. K-injectives in Grothendieck categories
    • 19.13. Additional remarks on Grothendieck abelian categories
    • 19.14. Other chapters
  • Chapter 20. Cohomology of Sheaves
    • 20.1. Introduction
    • 20.2. Topics
    • 20.3. Cohomology of sheaves
    • 20.4. Derived functors
    • 20.5. First cohomology and torsors
    • 20.6. First cohomology and extensions
    • 20.7. First cohomology and invertible sheaves
    • 20.8. Locality of cohomology
    • 20.9. Mayer-Vietoris
    • 20.10. The Cech complex and Cech cohomology
    • 20.11. Cech cohomology as a functor on presheaves
    • 20.12. Cech cohomology and cohomology
    • 20.13. Flasque sheaves
    • 20.14. The Leray spectral sequence
    • 20.15. Functoriality of cohomology
    • 20.16. Refinements and Cech cohomology
    • 20.17. Cohomology on Hausdorff quasi-compact spaces
    • 20.18. The base change map
    • 20.19. Proper base change in topology
    • 20.20. Cohomology and colimits
    • 20.21. Vanishing on Noetherian topological spaces
    • 20.22. Cohomology with support in a closed
    • 20.23. Cohomology on spectral spaces
    • 20.24. The alternating Cech complex
    • 20.25. Alternative view of the Cech complex
    • 20.26. Cech cohomology of complexes
    • 20.27. Flat resolutions
    • 20.28. Derived pullback
    • 20.29. Cohomology of unbounded complexes
    • 20.30. Unbounded Mayer-Vietoris
    • 20.31. Derived limits
    • 20.32. Producing K-injective resolutions
    • 20.33. Cech cohomology of unbounded complexes
    • 20.34. Hom complexes
    • 20.35. Internal hom in the derived category
    • 20.36. Ext sheaves
    • 20.37. Global derived hom
    • 20.38. Strictly perfect complexes
    • 20.39. Pseudo-coherent modules
    • 20.40. Tor dimension
    • 20.41. Perfect complexes
    • 20.42. Compact objects
    • 20.43. Projection formula
    • 20.44. Other chapters
  • Chapter 21. Cohomology on Sites
    • 21.1. Introduction
    • 21.2. Topics
    • 21.3. Cohomology of sheaves
    • 21.4. Derived functors
    • 21.5. First cohomology and torsors
    • 21.6. First cohomology and extensions
    • 21.7. First cohomology and invertible sheaves
    • 21.8. Locality of cohomology
    • 21.9. The Cech complex and Cech cohomology
    • 21.10. Cech cohomology as a functor on presheaves
    • 21.11. Cech cohomology and cohomology
    • 21.12. Cohomology of modules
    • 21.13. Limp sheaves
    • 21.14. The Leray spectral sequence
    • 21.15. The base change map
    • 21.16. Cohomology and colimits
    • 21.17. Flat resolutions
    • 21.18. Derived pullback
    • 21.19. Cohomology of unbounded complexes
    • 21.20. Some properties of K-injective complexes
    • 21.21. Derived and homotopy limits
    • 21.22. Producing K-injective resolutions
    • 21.23. Cohomology on Hausdorff and locally quasi-compact spaces
    • 21.24. Spectral sequences for Ext
    • 21.25. Hom complexes
    • 21.26. Internal hom in the derived category
    • 21.27. Global derived hom
    • 21.28. Derived lower shriek
    • 21.29. Derived lower shriek for fibred categories
    • 21.30. Homology on a category
    • 21.31. Calculating derived lower shriek
    • 21.32. Simplicial modules
    • 21.33. Cohomology on a category
    • 21.34. Strictly perfect complexes
    • 21.35. Pseudo-coherent modules
    • 21.36. Tor dimension
    • 21.37. Perfect complexes
    • 21.38. Projection formula
    • 21.39. Weakly contractible objects
    • 21.40. Compact objects
    • 21.41. Complexes with locally constant cohomology sheaves
    • 21.42. Other chapters
  • Chapter 22. Differential Graded Algebra
    • 22.1. Introduction
    • 22.2. Conventions
    • 22.3. Differential graded algebras
    • 22.4. Differential graded modules
    • 22.5. The homotopy category
    • 22.6. Cones
    • 22.7. Admissible short exact sequences
    • 22.8. Distinguished triangles
    • 22.9. Cones and distinguished triangles
    • 22.10. The homotopy category is triangulated
    • 22.11. Projective modules over algebras
    • 22.12. Injective modules over algebras
    • 22.13. P-resolutions
    • 22.14. I-resolutions
    • 22.15. The derived category
    • 22.16. The canonical delta-functor
    • 22.17. Linear categories
    • 22.18. Graded categories
    • 22.19. Differential graded categories
    • 22.20. Obtaining triangulated categories
    • 22.21. Derived Hom
    • 22.22. Variant of derived Hom
    • 22.23. Tensor product
    • 22.24. Derived tensor product
    • 22.25. Variant of derived tensor product
    • 22.26. Characterizing compact objects
    • 22.27. Equivalences of derived categories
    • 22.28. Other chapters
  • Chapter 23. Divided Power Algebra
    • 23.1. Introduction
    • 23.2. Divided powers
    • 23.3. Divided power rings
    • 23.4. Extending divided powers
    • 23.5. Divided power polynomial algebras
    • 23.6. Tate resolutions
    • 23.7. Application to complete intersections
    • 23.8. Local complete intersection rings
    • 23.9. Local complete intersection maps
    • 23.10. Other chapters
  • Chapter 24. Hypercoverings
    • 24.1. Introduction
    • 24.2. Hypercoverings
    • 24.3. Acyclicity
    • 24.4. Cech cohomology and hypercoverings
    • 24.5. Hypercoverings a la Verdier
    • 24.6. Covering hypercoverings
    • 24.7. Adding simplices
    • 24.8. Homotopies
    • 24.9. Cohomology and hypercoverings
    • 24.10. Hypercoverings of spaces
    • 24.11. Hypercoverings and weakly contractible objects
    • 24.12. Other chapters
  • Chapter 25. Schemes
    • 25.1. Introduction
    • 25.2. Locally ringed spaces
    • 25.3. Open immersions of locally ringed spaces
    • 25.4. Closed immersions of locally ringed spaces
    • 25.5. Affine schemes
    • 25.6. The category of affine schemes
    • 25.7. Quasi-coherent sheaves on affines
    • 25.8. Closed subspaces of affine schemes
    • 25.9. Schemes
    • 25.10. Immersions of schemes
    • 25.11. Zariski topology of schemes
    • 25.12. Reduced schemes
    • 25.13. Points of schemes
    • 25.14. Glueing schemes
    • 25.15. A representability criterion
    • 25.16. Existence of fibre products of schemes
    • 25.17. Fibre products of schemes
    • 25.18. Base change in algebraic geometry
    • 25.19. Quasi-compact morphisms
    • 25.20. Valuative criterion for universal closedness
    • 25.21. Separation axioms
    • 25.22. Valuative criterion of separatedness
    • 25.23. Monomorphisms
    • 25.24. Functoriality for quasi-coherent modules
    • 25.25. Other chapters
  • Chapter 26. Constructions of Schemes
    • 26.1. Introduction
    • 26.2. Relative glueing
    • 26.3. Relative spectrum via glueing
    • 26.4. Relative spectrum as a functor
    • 26.5. Affine n-space
    • 26.6. Vector bundles
    • 26.7. Cones
    • 26.8. Proj of a graded ring
    • 26.9. Quasi-coherent sheaves on Proj
    • 26.10. Invertible sheaves on Proj
    • 26.11. Functoriality of Proj
    • 26.12. Morphisms into Proj
    • 26.13. Projective space
    • 26.14. Invertible sheaves and morphisms into Proj
    • 26.15. Relative Proj via glueing
    • 26.16. Relative Proj as a functor
    • 26.17. Quasi-coherent sheaves on relative Proj
    • 26.18. Functoriality of relative Proj
    • 26.19. Invertible sheaves and morphisms into relative Proj
    • 26.20. Twisting by invertible sheaves and relative Proj
    • 26.21. Projective bundles
    • 26.22. Grassmannians
    • 26.23. Other chapters
  • Chapter 27. Properties of Schemes
    • 27.1. Introduction
    • 27.2. Constructible sets
    • 27.3. Integral, irreducible, and reduced schemes
    • 27.4. Types of schemes defined by properties of rings
    • 27.5. Noetherian schemes
    • 27.6. Jacobson schemes
    • 27.7. Normal schemes
    • 27.8. Cohen-Macaulay schemes
    • 27.9. Regular schemes
    • 27.10. Dimension
    • 27.11. Catenary schemes
    • 27.12. Serre's conditions
    • 27.13. Japanese and Nagata schemes
    • 27.14. The singular locus
    • 27.15. Local irreducibility
    • 27.16. Characterizing modules of finite type and finite presentation
    • 27.17. Sections over principal opens
    • 27.18. Quasi-affine schemes
    • 27.19. Flat modules
    • 27.20. Locally free modules
    • 27.21. Locally projective modules
    • 27.22. Extending quasi-coherent sheaves
    • 27.23. Gabber's result
    • 27.24. Sections with support in a closed subset
    • 27.25. Sections of quasi-coherent sheaves
    • 27.26. Ample invertible sheaves
    • 27.27. Affine and quasi-affine schemes
    • 27.28. Quasi-coherent sheaves and ample invertible sheaves
    • 27.29. Finding suitable affine opens
    • 27.30. Other chapters
  • Chapter 28. Morphisms of Schemes
    • 28.1. Introduction
    • 28.2. Closed immersions
    • 28.3. Immersions
    • 28.4. Closed immersions and quasi-coherent sheaves
    • 28.5. Supports of modules
    • 28.6. Scheme theoretic image
    • 28.7. Scheme theoretic closure and density
    • 28.8. Dominant morphisms
    • 28.9. Rational maps
    • 28.10. Surjective morphisms
    • 28.11. Radicial and universally injective morphisms
    • 28.12. Affine morphisms
    • 28.13. Quasi-affine morphisms
    • 28.14. Types of morphisms defined by properties of ring maps
    • 28.15. Morphisms of finite type
    • 28.16. Points of finite type and Jacobson schemes
    • 28.17. Universally catenary schemes
    • 28.18. Nagata schemes, reprise
    • 28.19. The singular locus, reprise
    • 28.20. Quasi-finite morphisms
    • 28.21. Morphisms of finite presentation
    • 28.22. Constructible sets
    • 28.23. Open morphisms
    • 28.24. Submersive morphisms
    • 28.25. Flat morphisms
    • 28.26. Flat closed immersions
    • 28.27. Generic flatness
    • 28.28. Morphisms and dimensions of fibres
    • 28.29. Morphisms of given relative dimension
    • 28.30. The dimension formula
    • 28.31. Syntomic morphisms
    • 28.32. Conormal sheaf of an immersion
    • 28.33. Sheaf of differentials of a morphism
    • 28.34. Smooth morphisms
    • 28.35. Unramified morphisms
    • 28.36. Étale morphisms
    • 28.37. Relatively ample sheaves
    • 28.38. Very ample sheaves
    • 28.39. Ample and very ample sheaves relative to finite type morphisms
    • 28.40. Quasi-projective morphisms
    • 28.41. Proper morphisms
    • 28.42. Projective morphisms
    • 28.43. Integral and finite morphisms
    • 28.44. Universal homeomorphisms
    • 28.45. Finite locally free morphisms
    • 28.46. Birational morphisms
    • 28.47. Generically finite morphisms
    • 28.48. Relative normalization
    • 28.49. Normalization
    • 28.50. Zariski's Main Theorem (algebraic version)
    • 28.51. Universally bounded fibres
    • 28.52. Other chapters
  • Chapter 29. Cohomology of Schemes
    • 29.1. Introduction
    • 29.2. Cech cohomology of quasi-coherent sheaves
    • 29.3. Vanishing of cohomology
    • 29.4. Quasi-coherence of higher direct images
    • 29.5. Cohomology and base change, I
    • 29.6. Colimits and higher direct images
    • 29.7. Cohomology and base change, II
    • 29.8. Cohomology of projective space
    • 29.9. Coherent sheaves on locally Noetherian schemes
    • 29.10. Coherent sheaves on Noetherian schemes
    • 29.11. Depth
    • 29.12. Devissage of coherent sheaves
    • 29.13. Finite morphisms and affines
    • 29.14. Coherent sheaves on Proj
    • 29.15. Higher direct images along projective morphisms
    • 29.16. Ample invertible sheaves and cohomology
    • 29.17. Chow's Lemma
    • 29.18. Higher direct images of coherent sheaves
    • 29.19. The theorem on formal functions
    • 29.20. Applications of the theorem on formal functions
    • 29.21. Cohomology and base change, III
    • 29.22. Grothendieck's existence theorem, I
    • 29.23. Grothendieck's existence theorem, II
    • 29.24. Grothendieck's algebraization theorem
    • 29.25. Other chapters
  • Chapter 30. Divisors
    • 30.1. Introduction
    • 30.2. Associated points
    • 30.3. Morphisms and associated points
    • 30.4. Embedded points
    • 30.5. Weakly associated points
    • 30.6. Morphisms and weakly associated points
    • 30.7. Relative assassin
    • 30.8. Relative weak assassin
    • 30.9. Torsion free modules
    • 30.10. Reflexive modules
    • 30.11. Effective Cartier divisors
    • 30.12. Effective Cartier divisors on Noetherian schemes
    • 30.13. Complements of affine opens
    • 30.14. Norms
    • 30.15. Relative effective Cartier divisors
    • 30.16. The normal cone of an immersion
    • 30.17. Regular ideal sheaves
    • 30.18. Regular immersions
    • 30.19. Relative regular immersions
    • 30.20. Meromorphic functions and sections
    • 30.21. Weil divisors
    • 30.22. The Weil divisor class associated to an invertible module
    • 30.23. More on invertible modules
    • 30.24. Relative Proj
    • 30.25. Closed subschemes of relative proj
    • 30.26. Blowing up
    • 30.27. Strict transform
    • 30.28. Admissible blowups
    • 30.29. Modifications
    • 30.30. Other chapters
  • Chapter 31. Limits of Schemes
    • 31.1. Introduction
    • 31.2. Directed limits of schemes with affine transition maps
    • 31.3. Descending properties
    • 31.4. Absolute Noetherian Approximation
    • 31.5. Limits and morphisms of finite presentation
    • 31.6. Relative approximation
    • 31.7. Descending properties of morphisms
    • 31.8. Finite type closed in finite presentation
    • 31.9. Descending relative objects
    • 31.10. Characterizing affine schemes
    • 31.11. Variants of Chow's Lemma
    • 31.12. Applications of Chow's lemma
    • 31.13. Universally closed morphisms
    • 31.14. Limits and dimensions of fibres
    • 31.15. Application to modifications
    • 31.16. Other chapters
  • Chapter 32. Varieties
    • 32.1. Introduction
    • 32.2. Notation
    • 32.3. Varieties
    • 32.4. Geometrically reduced schemes
    • 32.5. Geometrically connected schemes
    • 32.6. Geometrically irreducible schemes
    • 32.7. Geometrically integral schemes
    • 32.8. Geometrically normal schemes
    • 32.9. Change of fields and locally Noetherian schemes
    • 32.10. Geometrically regular schemes
    • 32.11. Change of fields and the Cohen-Macaulay property
    • 32.12. Change of fields and the Jacobson property
    • 32.13. Change of fields and ample invertible sheaves
    • 32.14. Tangent spaces
    • 32.15. Generically finite morphisms
    • 32.16. Dimension of fibres
    • 32.17. Algebraic schemes
    • 32.18. Global generation
    • 32.19. Closures of products
    • 32.20. Schemes smooth over fields
    • 32.21. Types of varieties
    • 32.22. Groups of invertible functions
    • 32.23. Künneth formula
    • 32.24. Picard groups of varieties
    • 32.25. Uniqueness of base field
    • 32.26. Euler characteristics
    • 32.27. Projective space
    • 32.28. Coherent sheaves on projective space
    • 32.29. Glueing dimension one rings
    • 32.30. One dimensional Noetherian schemes
    • 32.31. Finding affine opens
    • 32.32. Curves
    • 32.33. Degrees on curves
    • 32.34. Numerical intersections
    • 32.35. Other chapters
  • Chapter 33. Topologies on Schemes
    • 33.1. Introduction
    • 33.2. The general procedure
    • 33.3. The Zariski topology
    • 33.4. The étale topology
    • 33.5. The smooth topology
    • 33.6. The syntomic topology
    • 33.7. The fppf topology
    • 33.8. The fpqc topology
    • 33.9. Change of topologies
    • 33.10. Change of big sites
    • 33.11. Other chapters
  • Chapter 34. Descent
    • 34.1. Introduction
    • 34.2. Descent data for quasi-coherent sheaves
    • 34.3. Descent for modules
    • 34.4. Descent for universally injective morphisms
    • 34.5. Fpqc descent of quasi-coherent sheaves
    • 34.6. Descent of finiteness properties of modules
    • 34.7. Quasi-coherent sheaves and topologies
    • 34.8. Parasitic modules
    • 34.9. Fpqc coverings are universal effective epimorphisms
    • 34.10. Descent of finiteness properties of morphisms
    • 34.11. Local properties of schemes
    • 34.12. Properties of schemes local in the fppf topology
    • 34.13. Properties of schemes local in the syntomic topology
    • 34.14. Properties of schemes local in the smooth topology
    • 34.15. Variants on descending properties
    • 34.16. Germs of schemes
    • 34.17. Local properties of germs
    • 34.18. Properties of morphisms local on the target
    • 34.19. Properties of morphisms local in the fpqc topology on the target
    • 34.20. Properties of morphisms local in the fppf topology on the target
    • 34.21. Application of fpqc descent of properties of morphisms
    • 34.22. Properties of morphisms local on the source
    • 34.23. Properties of morphisms local in the fpqc topology on the source
    • 34.24. Properties of morphisms local in the fppf topology on the source
    • 34.25. Properties of morphisms local in the syntomic topology on the source
    • 34.26. Properties of morphisms local in the smooth topology on the source
    • 34.27. Properties of morphisms local in the étale topology on the source
    • 34.28. Properties of morphisms étale local on source-and-target
    • 34.29. Properties of morphisms of germs local on source-and-target
    • 34.30. Descent data for schemes over schemes
    • 34.31. Fully faithfulness of the pullback functors
    • 34.32. Descending types of morphisms
    • 34.33. Descending affine morphisms
    • 34.34. Descending quasi-affine morphisms
    • 34.35. Descent data in terms of sheaves
    • 34.36. Other chapters
  • Chapter 35. Derived Categories of Schemes
    • 35.1. Introduction
    • 35.2. Conventions
    • 35.3. Derived category of quasi-coherent modules
    • 35.4. Total direct image
    • 35.5. Affine morphisms
    • 35.6. Derived category of coherent modules
    • 35.7. The coherator
    • 35.8. The coherator for Noetherian schemes
    • 35.9. Koszul complexes
    • 35.10. Pseudo-coherent and perfect complexes
    • 35.11. Descent finiteness properties of complexes
    • 35.12. Lifting complexes
    • 35.13. Approximation by perfect complexes
    • 35.14. Generating derived categories
    • 35.15. An example generator
    • 35.16. Compact and perfect objects
    • 35.17. Derived categories as module categories
    • 35.18. Cohomology and base change, IV
    • 35.19. Producing perfect complexes
    • 35.20. Cohomology, Ext groups, and base change
    • 35.21. Limits and derived categories
    • 35.22. Cohomology and base change, V
    • 35.23. Perfect complexes
    • 35.24. Applications
    • 35.25. Theorem of the cube
    • 35.26. Formal functions for a principal ideal
    • 35.27. Other chapters
  • Chapter 36. More on Morphisms
    • 36.1. Introduction
    • 36.2. Thickenings
    • 36.3. First order infinitesimal neighbourhood
    • 36.4. Formally unramified morphisms
    • 36.5. Universal first order thickenings
    • 36.6. Formally étale morphisms
    • 36.7. Infinitesimal deformations of maps
    • 36.8. Infinitesimal deformations of schemes
    • 36.9. Formally smooth morphisms
    • 36.10. Smoothness over a Noetherian base
    • 36.11. Pushouts in the category of schemes
    • 36.12. Openness of the flat locus
    • 36.13. Critère de platitude par fibres
    • 36.14. Normalization revisited
    • 36.15. Normal morphisms
    • 36.16. Regular morphisms
    • 36.17. Cohen-Macaulay morphisms
    • 36.18. Slicing Cohen-Macaulay morphisms
    • 36.19. Generic fibres
    • 36.20. Relative assassins
    • 36.21. Reduced fibres
    • 36.22. Irreducible components of fibres
    • 36.23. Connected components of fibres
    • 36.24. Connected components meeting a section
    • 36.25. Dimension of fibres
    • 36.26. Limit arguments
    • 36.27. Étale neighbourhoods
    • 36.28. Slicing smooth morphisms
    • 36.29. Finite free locally dominates étale
    • 36.30. Étale localization of quasi-finite morphisms
    • 36.31. Zariski's Main Theorem
    • 36.32. Application to morphisms with connected fibres
    • 36.33. Application to the structure of finite type morphisms
    • 36.34. Application to the fppf topology
    • 36.35. Quasi-projective schemes
    • 36.36. Projective schemes
    • 36.37. Closed points in fibres
    • 36.38. Stein factorization
    • 36.39. Descending separated locally quasi-finite morphisms
    • 36.40. Relative finite presentation
    • 36.41. Relative pseudo-coherence
    • 36.42. Pseudo-coherent morphisms
    • 36.43. Perfect morphisms
    • 36.44. Local complete intersection morphisms
    • 36.45. Exact sequences of differentials and conormal sheaves
    • 36.46. Weakly étale morphisms
    • 36.47. Reduced fibre theorem
    • 36.48. Ind-quasi-affine morphisms
    • 36.49. Relative morphisms
    • 36.50. Other chapters
  • Chapter 37. More on Flatness
    • 37.1. Introduction
    • 37.2. Lemmas on étale localization
    • 37.3. The local structure of a finite type module
    • 37.4. One step dévissage
    • 37.5. Complete dévissage
    • 37.6. Translation into algebra
    • 37.7. Localization and universally injective maps
    • 37.8. Completion and Mittag-Leffler modules
    • 37.9. Projective modules
    • 37.10. Flat finite type modules, Part I
    • 37.11. Extending properties from an open
    • 37.12. Flat finitely presented modules
    • 37.13. Flat finite type modules, Part II
    • 37.14. Examples of relatively pure modules
    • 37.15. Impurities
    • 37.16. Relatively pure modules
    • 37.17. Examples of relatively pure sheaves
    • 37.18. A criterion for purity
    • 37.19. How purity is used
    • 37.20. Flattening functors
    • 37.21. Flattening stratifications
    • 37.22. Flattening stratification over an Artinian ring
    • 37.23. Flattening a map
    • 37.24. Flattening in the local case
    • 37.25. Variants of a lemma
    • 37.26. Flat finite type modules, Part III
    • 37.27. Universal flattening
    • 37.28. Blowing up and flatness
    • 37.29. Applications
    • 37.30. Other chapters
  • Chapter 38. Groupoid Schemes
    • 38.1. Introduction
    • 38.2. Notation
    • 38.3. Equivalence relations
    • 38.4. Group schemes
    • 38.5. Examples of group schemes
    • 38.6. Properties of group schemes
    • 38.7. Properties of group schemes over a field
    • 38.8. Properties of algebraic group schemes
    • 38.9. Abelian varieties
    • 38.10. Actions of group schemes
    • 38.11. Principal homogeneous spaces
    • 38.12. Equivariant quasi-coherent sheaves
    • 38.13. Groupoids
    • 38.14. Quasi-coherent sheaves on groupoids
    • 38.15. Colimits of quasi-coherent modules
    • 38.16. Groupoids and group schemes
    • 38.17. The stabilizer group scheme
    • 38.18. Restricting groupoids
    • 38.19. Invariant subschemes
    • 38.20. Quotient sheaves
    • 38.21. Descent in terms of groupoids
    • 38.22. Separation conditions
    • 38.23. Finite flat groupoids, affine case
    • 38.24. Finite flat groupoids
    • 38.25. Other chapters
  • Chapter 39. More on Groupoid Schemes
    • 39.1. Introduction
    • 39.2. Notation
    • 39.3. Useful diagrams
    • 39.4. Sheaf of differentials
    • 39.5. Properties of groupoids
    • 39.6. Comparing fibres
    • 39.7. Cohen-Macaulay presentations
    • 39.8. Restricting groupoids
    • 39.9. Properties of groupoids on fields
    • 39.10. Morphisms of groupoids on fields
    • 39.11. Slicing groupoids
    • 39.12. Étale localization of groupoids
    • 39.13. Finite groupoids
    • 39.14. Descending ind-quasi-affine morphisms
    • 39.15. Other chapters
  • Chapter 40. Étale Morphisms of Schemes
    • 40.1. Introduction
    • 40.2. Conventions
    • 40.3. Unramified morphisms
    • 40.4. Three other characterizations of unramified morphisms
    • 40.5. The functorial characterization of unramified morphisms
    • 40.6. Topological properties of unramified morphisms
    • 40.7. Universally injective, unramified morphisms
    • 40.8. Examples of unramified morphisms
    • 40.9. Flat morphisms
    • 40.10. Topological properties of flat morphisms
    • 40.11. Étale morphisms
    • 40.12. The structure theorem
    • 40.13. Étale and smooth morphisms
    • 40.14. Topological properties of étale morphisms
    • 40.15. Topological invariance of the étale topology
    • 40.16. The functorial characterization
    • 40.17. Étale local structure of unramified morphisms
    • 40.18. Étale local structure of étale morphisms
    • 40.19. Permanence properties
    • 40.20. Other chapters
  • Chapter 41. Chow Homology and Chern Classes
    • 41.1. Introduction
    • 41.2. Determinants of finite length modules
    • 41.3. Periodic complexes and Herbrand quotients
    • 41.4. Periodic complexes and determinants
    • 41.5. Symbols
    • 41.6. Lengths and determinants
    • 41.7. Application to tame symbol
    • 41.8. Setup
    • 41.9. Cycles
    • 41.10. Cycle associated to a closed subscheme
    • 41.11. Cycle associated to a coherent sheaf
    • 41.12. Preparation for proper pushforward
    • 41.13. Proper pushforward
    • 41.14. Preparation for flat pullback
    • 41.15. Flat pullback
    • 41.16. Push and pull
    • 41.17. Preparation for principal divisors
    • 41.18. Principal divisors
    • 41.19. Principal divisors and pushforward
    • 41.20. Rational equivalence
    • 41.21. Rational equivalence and push and pull
    • 41.22. Rational equivalence and the projective line
    • 41.23. The divisor associated to an invertible sheaf
    • 41.24. Intersecting with an invertible sheaf
    • 41.25. Intersecting with an invertible sheaf and push and pull
    • 41.26. The key formula
    • 41.27. Intersecting with an invertible sheaf and rational equivalence
    • 41.28. Intersecting with effective Cartier divisors
    • 41.29. Gysin homomorphisms
    • 41.30. Relative effective Cartier divisors
    • 41.31. Affine bundles
    • 41.32. Bivariant intersection theory
    • 41.33. Projective space bundle formula
    • 41.34. The Chern classes of a vector bundle
    • 41.35. Intersecting with chern classes
    • 41.36. Polynomial relations among chern classes
    • 41.37. Additivity of chern classes
    • 41.38. The splitting principle
    • 41.39. Chern classes and tensor product
    • 41.40. Todd classes
    • 41.41. Degrees of zero cycles
    • 41.42. Grothendieck-Riemann-Roch
    • 41.43. Appendix
    • 41.44. Other chapters
  • Chapter 42. Intersection Theory
    • 42.1. Introduction
    • 42.2. Conventions
    • 42.3. Cycles
    • 42.4. Cycle associated to closed subscheme
    • 42.5. Cycle associated to a coherent sheaf
    • 42.6. Proper pushforward
    • 42.7. Flat pullback
    • 42.8. Rational Equivalence
    • 42.9. Rational equivalence and rational functions
    • 42.10. Proper pushforward and rational equivalence
    • 42.11. Flat pullback and rational equivalence
    • 42.12. The short exact sequence for an open
    • 42.13. Proper intersections
    • 42.14. Intersection multiplicities using Tor formula
    • 42.15. Algebraic multiplicities
    • 42.16. Computing intersection multiplicities
    • 42.17. Intersection product using Tor formula
    • 42.18. Exterior product
    • 42.19. Reduction to the diagonal
    • 42.20. Associativity of intersections
    • 42.21. Flat pullback and intersection products
    • 42.22. Projection formula for flat proper morphisms
    • 42.23. Projections
    • 42.24. Moving Lemma
    • 42.25. Intersection products and rational equivalence
    • 42.26. Chow rings
    • 42.27. Pullback for a general morphism
    • 42.28. Pullback of cycles
    • 42.29. Other chapters
  • Chapter 43. Picard Schemes of Curves
    • 43.1. Introduction
    • 43.2. Hilbert scheme of points
    • 43.3. Moduli of divisors on smooth curves
    • 43.4. The Picard functor
    • 43.5. A representability criterion
    • 43.6. The Picard scheme of a curve
    • 43.7. Other chapters
  • Chapter 44. Adequate Modules
    • 44.1. Introduction
    • 44.2. Conventions
    • 44.3. Adequate functors
    • 44.4. Higher exts of adequate functors
    • 44.5. Adequate modules
    • 44.6. Parasitic adequate modules
    • 44.7. Derived categories of adequate modules, I
    • 44.8. Pure extensions
    • 44.9. Higher exts of quasi-coherent sheaves on the big site
    • 44.10. Derived categories of adequate modules, II
    • 44.11. Other chapters
  • Chapter 45. Dualizing Complexes
    • 45.1. Introduction
    • 45.2. Essential surjections and injections
    • 45.3. Injective modules
    • 45.4. Projective covers
    • 45.5. Injective hulls
    • 45.6. Duality over Artinian local rings
    • 45.7. Injective hull of the residue field
    • 45.8. Deriving torsion
    • 45.9. Local cohomology
    • 45.10. Local cohomology for Noetherian rings
    • 45.11. Depth
    • 45.12. Torsion versus complete modules
    • 45.13. Formally catenary rings
    • 45.14. Finiteness of local cohomology, I
    • 45.15. Finiteness of pushforwards, I
    • 45.16. Trivial duality for a ring map
    • 45.17. Dualizing complexes
    • 45.18. Dualizing complexes over local rings
    • 45.19. The dimension function of a dualizing complex
    • 45.20. The local duality theorem
    • 45.21. Dualizing complexes on schemes
    • 45.22. Right adjoint of pushforward
    • 45.23. Right adjoint of pushforward and base change
    • 45.24. Right adjoint of pushforward and trace maps
    • 45.25. Right adjoint of pushforward and pullback
    • 45.26. Right adjoint of pushforward for closed immersions
    • 45.27. Right adjoint of pushforward for finite morphisms
    • 45.28. Right adjoint of pushforward for perfect proper morphisms
    • 45.29. Right adjoint of pushforward for effective Cartier divisors
    • 45.30. Right adjoint of pushforward in examples
    • 45.31. Compactifications
    • 45.32. Upper shriek functors
    • 45.33. Properties of upper shriek functors
    • 45.34. A duality theory
    • 45.35. Glueing dualizing complexes
    • 45.36. Dualizing modules
    • 45.37. Cohen-Macaulay schemes
    • 45.38. Gorenstein schemes
    • 45.39. Formal fibres
    • 45.40. Finiteness of local cohomology, II
    • 45.41. Finiteness of pushforwards, II
    • 45.42. Other chapters
  • Chapter 46. Algebraic Curves
    • 46.1. Introduction
    • 46.2. Riemann-Roch and duality
    • 46.3. Some vanishing results
    • 46.4. Other chapters
  • Chapter 47. Resolution of Surfaces
    • 47.1. Introduction
    • 47.2. A trace map in positive characteristic
    • 47.3. Quadratic transformations
    • 47.4. Dominating by quadratic transformations
    • 47.5. Dominating by normalized blowups
    • 47.6. Modifying over local rings
    • 47.7. Vanishing
    • 47.8. Boundedness
    • 47.9. Rational singularities
    • 47.10. Formal arcs
    • 47.11. Base change to the completion
    • 47.12. Rational double points
    • 47.13. Implied properties
    • 47.14. Resolution
    • 47.15. Embedded resolution
    • 47.16. Other chapters
  • Chapter 48. Fundamental Groups of Schemes
    • 48.1. Introduction
    • 48.2. Schemes étale over a point
    • 48.3. Galois categories
    • 48.4. Finite étale morphisms
    • 48.5. Fundamental groups
    • 48.6. Finite étale covers of proper schemes
    • 48.7. Local connectedness
    • 48.8. Fundamental groups of normal schemes
    • 48.9. Finite étale covers of punctured spectra, I
    • 48.10. Purity in local case, I
    • 48.11. Purity of branch locus
    • 48.12. Finite étale covers of punctured spectra, II
    • 48.13. Purity in local case, II
    • 48.14. Ramification theory
    • 48.15. Tame ramification
    • 48.16. Other chapters
  • Chapter 49. Étale Cohomology
    • 49.1. Introduction
    • 49.2. Which sections to skip on a first reading?
    • 49.3. Prologue
    • 49.4. The étale topology
    • 49.5. Feats of the étale topology
    • 49.6. A computation
    • 49.7. Nontorsion coefficients
    • 49.8. Sheaf theory
    • 49.9. Presheaves
    • 49.10. Sites
    • 49.11. Sheaves
    • 49.12. The example of G-sets
    • 49.13. Sheafification
    • 49.14. Cohomology
    • 49.15. The fpqc topology
    • 49.16. Faithfully flat descent
    • 49.17. Quasi-coherent sheaves
    • 49.18. Cech cohomology
    • 49.19. The Cech-to-cohomology spectral sequence
    • 49.20. Big and small sites of schemes
    • 49.21. The étale topos
    • 49.22. Cohomology of quasi-coherent sheaves
    • 49.23. Examples of sheaves
    • 49.24. Picard groups
    • 49.25. The étale site
    • 49.26. Étale morphisms
    • 49.27. Étale coverings
    • 49.28. Kummer theory
    • 49.29. Neighborhoods, stalks and points
    • 49.30. Points in other topologies
    • 49.31. Supports of abelian sheaves
    • 49.32. Henselian rings
    • 49.33. Stalks of the structure sheaf
    • 49.34. Functoriality of small étale topos
    • 49.35. Direct images
    • 49.36. Inverse image
    • 49.37. Functoriality of big topoi
    • 49.38. Functoriality and sheaves of modules
    • 49.39. Comparing big and small topoi
    • 49.40. Comparing topologies
    • 49.41. Recovering morphisms
    • 49.42. Push and pull
    • 49.43. Property (A)
    • 49.44. Property (B)
    • 49.45. Property (C)
    • 49.46. Topological invariance of the small étale site
    • 49.47. Closed immersions and pushforward
    • 49.48. Integral universally injective morphisms
    • 49.49. Big sites and pushforward
    • 49.50. Exactness of big lower shriek
    • 49.51. Étale cohomology
    • 49.52. Colimits
    • 49.53. Stalks of higher direct images
    • 49.54. The Leray spectral sequence
    • 49.55. Vanishing of finite higher direct images
    • 49.56. Galois action on stalks
    • 49.57. Group cohomology
    • 49.58. Cohomology of a point
    • 49.59. Cohomology of curves
    • 49.60. Brauer groups
    • 49.61. The Brauer group of a scheme
    • 49.62. Galois cohomology
    • 49.63. Higher vanishing for the multiplicative group
    • 49.64. The Artin-Schreier sequence
    • 49.65. Picard groups of curves
    • 49.66. Extension by zero
    • 49.67. Locally constant sheaves
    • 49.68. Constructible sheaves
    • 49.69. Auxiliary lemmas on morphisms
    • 49.70. More on constructible sheaves
    • 49.71. Constructible sheaves on Noetherian schemes
    • 49.72. Cohomology with support in a closed subscheme
    • 49.73. Affine analog of proper base change
    • 49.74. Cohomology of torsion sheaves on curves
    • 49.75. First cohomology of proper schemes
    • 49.76. The proper base change theorem
    • 49.77. Applications of proper base change
    • 49.78. The trace formula
    • 49.79. Frobenii
    • 49.80. Traces
    • 49.81. Why derived categories?
    • 49.82. Derived categories
    • 49.83. Filtered derived category
    • 49.84. Filtered derived functors
    • 49.85. Application of filtered complexes
    • 49.86. Perfectness
    • 49.87. Filtrations and perfect complexes
    • 49.88. Characterizing perfect objects
    • 49.89. Complexes with constructible cohomology
    • 49.90. Cohomology of nice complexes
    • 49.91. Lefschetz numbers
    • 49.92. Preliminaries and sorites
    • 49.93. Proof of the trace formula
    • 49.94. Applications
    • 49.95. On l-adic sheaves
    • 49.96. L-functions
    • 49.97. Cohomological interpretation
    • 49.98. List of things which we should add above
    • 49.99. Examples of L-functions
    • 49.100. Constant sheaves
    • 49.101. The Legendre family
    • 49.102. Exponential sums
    • 49.103. Trace formula in terms of fundamental groups
    • 49.104. Fundamental groups
    • 49.105. Profinite groups, cohomology and homology
    • 49.106. Cohomology of curves, revisited
    • 49.107. Abstract trace formula
    • 49.108. Automorphic forms and sheaves
    • 49.109. Counting points
    • 49.110. Precise form of Chebotarev
    • 49.111. How many primes decompose completely?
    • 49.112. How many points are there really?
    • 49.113. Other chapters
  • Chapter 50. Crystalline Cohomology
    • 50.1. Introduction
    • 50.2. Divided power envelope
    • 50.3. Some explicit divided power thickenings
    • 50.4. Compatibility
    • 50.5. Affine crystalline site
    • 50.6. Module of differentials
    • 50.7. Divided power schemes
    • 50.8. The big crystalline site
    • 50.9. The crystalline site
    • 50.10. Sheaves on the crystalline site
    • 50.11. Crystals in modules
    • 50.12. Sheaf of differentials
    • 50.13. Two universal thickenings
    • 50.14. The de Rham complex
    • 50.15. Connections
    • 50.16. Cosimplicial algebra
    • 50.17. Crystals in quasi-coherent modules
    • 50.18. General remarks on cohomology
    • 50.19. Cosimplicial preparations
    • 50.20. Divided power Poincaré lemma
    • 50.21. Cohomology in the affine case
    • 50.22. Two counter examples
    • 50.23. Applications
    • 50.24. Some further results
    • 50.25. Pulling back along purely inseparable maps
    • 50.26. Frobenius action on crystalline cohomology
    • 50.27. Other chapters
  • Chapter 51. Pro-étale Cohomology
    • 51.1. Introduction
    • 51.2. Some topology
    • 51.3. Local isomorphisms
    • 51.4. Ind-Zariski algebra
    • 51.5. Constructing w-local affine schemes
    • 51.6. Identifying local rings versus ind-Zariski
    • 51.7. Ind-étale algebra
    • 51.8. Constructing ind-étale algebras
    • 51.9. Weakly étale versus pro-étale
    • 51.10. Constructing w-contractible covers
    • 51.11. The pro-étale site
    • 51.12. Points of the pro-étale site
    • 51.13. Compact generation
    • 51.14. Generalities on derived completion
    • 51.15. Application to theorem on formal functions
    • 51.16. Derived completion in the constant Noetherian case
    • 51.17. Derived completion on the pro-étale site
    • 51.18. Comparison with the étale site
    • 51.19. Cohomology of a point
    • 51.20. Weakly contractible hypercoverings
    • 51.21. Functoriality of the pro-étale site
    • 51.22. Finite morphisms and pro-étale sites
    • 51.23. Closed immersions and pro-étale sites
    • 51.24. Extension by zero
    • 51.25. Constructible sheaves on the pro-étale site
    • 51.26. Constructible adic sheaves
    • 51.27. A suitable derived category
    • 51.28. Proper base change
    • 51.29. Other chapters
  • Chapter 52. Algebraic Spaces
    • 52.1. Introduction
    • 52.2. General remarks
    • 52.3. Representable morphisms of presheaves
    • 52.4. Lists of useful properties of morphisms of schemes
    • 52.5. Properties of representable morphisms of presheaves
    • 52.6. Algebraic spaces
    • 52.7. Fibre products of algebraic spaces
    • 52.8. Glueing algebraic spaces
    • 52.9. Presentations of algebraic spaces
    • 52.10. Algebraic spaces and equivalence relations
    • 52.11. Algebraic spaces, retrofitted
    • 52.12. Immersions and Zariski coverings of algebraic spaces
    • 52.13. Separation conditions on algebraic spaces
    • 52.14. Examples of algebraic spaces
    • 52.15. Change of big site
    • 52.16. Change of base scheme
    • 52.17. Other chapters
  • Chapter 53. Properties of Algebraic Spaces
    • 53.1. Introduction
    • 53.2. Conventions
    • 53.3. Separation axioms
    • 53.4. Points of algebraic spaces
    • 53.5. Quasi-compact spaces
    • 53.6. Special coverings
    • 53.7. Properties of Spaces defined by properties of schemes
    • 53.8. Dimension at a point
    • 53.9. Dimension of local rings
    • 53.10. Generic points
    • 53.11. Reduced spaces
    • 53.12. The schematic locus
    • 53.13. Obtaining a scheme
    • 53.14. Points on quasi-separated spaces
    • 53.15. Étale morphisms of algebraic spaces
    • 53.16. Spaces and fpqc coverings
    • 53.17. The étale site of an algebraic space
    • 53.18. Points of the small étale site
    • 53.19. Supports of abelian sheaves
    • 53.20. The structure sheaf of an algebraic space
    • 53.21. Stalks of the structure sheaf
    • 53.22. Local irreducibility
    • 53.23. Noetherian spaces
    • 53.24. Regular algebraic spaces
    • 53.25. Sheaves of modules on algebraic spaces
    • 53.26. Étale localization
    • 53.27. Recovering morphisms
    • 53.28. Quasi-coherent sheaves on algebraic spaces
    • 53.29. Properties of modules
    • 53.30. Locally projective modules
    • 53.31. Quasi-coherent sheaves and presentations
    • 53.32. Morphisms towards schemes
    • 53.33. Quotients by free actions
    • 53.34. Other chapters
  • Chapter 54. Morphisms of Algebraic Spaces
    • 54.1. Introduction
    • 54.2. Conventions
    • 54.3. Properties of representable morphisms
    • 54.4. Separation axioms
    • 54.5. Surjective morphisms
    • 54.6. Open morphisms
    • 54.7. Submersive morphisms
    • 54.8. Quasi-compact morphisms
    • 54.9. Universally closed morphisms
    • 54.10. Monomorphisms
    • 54.11. Pushforward of quasi-coherent sheaves
    • 54.12. Immersions
    • 54.13. Closed immersions
    • 54.14. Closed immersions and quasi-coherent sheaves
    • 54.15. Supports of modules
    • 54.16. Scheme theoretic image
    • 54.17. Scheme theoretic closure and density
    • 54.18. Dominant morphisms
    • 54.19. Universally injective morphisms
    • 54.20. Affine morphisms
    • 54.21. Quasi-affine morphisms
    • 54.22. Types of morphisms étale local on source-and-target
    • 54.23. Morphisms of finite type
    • 54.24. Points and geometric points
    • 54.25. Points of finite type
    • 54.26. Nagata spaces
    • 54.27. Quasi-finite morphisms
    • 54.28. Morphisms of finite presentation
    • 54.29. Flat morphisms
    • 54.30. Flat modules
    • 54.31. Generic flatness
    • 54.32. Relative dimension
    • 54.33. Morphisms and dimensions of fibres
    • 54.34. The dimension formula
    • 54.35. Syntomic morphisms
    • 54.36. Smooth morphisms
    • 54.37. Unramified morphisms
    • 54.38. Étale morphisms
    • 54.39. Proper morphisms
    • 54.40. Valuative criteria
    • 54.41. Valuative criterion for universal closedness
    • 54.42. Valuative criterion of separatedness
    • 54.43. Integral and finite morphisms
    • 54.44. Finite locally free morphisms
    • 54.45. Relative normalization of algebraic spaces
    • 54.46. Normalization
    • 54.47. Separated, locally quasi-finite morphisms
    • 54.48. Applications
    • 54.49. Zariski's Main Theorem (representable case)
    • 54.50. Universal homeomorphisms
    • 54.51. Other chapters
  • Chapter 55. Decent Algebraic Spaces
    • 55.1. Introduction
    • 55.2. Conventions
    • 55.3. Universally bounded fibres
    • 55.4. Finiteness conditions and points
    • 55.5. Conditions on algebraic spaces
    • 55.6. Reasonable and decent algebraic spaces
    • 55.7. Points and specializations
    • 55.8. Stratifying algebraic spaces by schemes
    • 55.9. Schematic locus
    • 55.10. Points on spaces
    • 55.11. Reduced singleton spaces
    • 55.12. Decent spaces
    • 55.13. Locally separated spaces
    • 55.14. Valuative criterion
    • 55.15. Relative conditions
    • 55.16. Points of fibres
    • 55.17. Monomorphisms
    • 55.18. Generic points
    • 55.19. Generically finite morphisms
    • 55.20. Birational morphisms
    • 55.21. Jacobson spaces
    • 55.22. Other chapters
  • Chapter 56. Cohomology of Algebraic Spaces
    • 56.1. Introduction
    • 56.2. Conventions
    • 56.3. Higher direct images
    • 56.4. Colimits and cohomology
    • 56.5. The alternating Cech complex
    • 56.6. Higher vanishing for quasi-coherent sheaves
    • 56.7. Vanishing for higher direct images
    • 56.8. Cohomology with support in a closed subspace
    • 56.9. Vanishing above the dimension
    • 56.10. Cohomology and base change, I
    • 56.11. Coherent modules on locally Noetherian algebraic spaces
    • 56.12. Coherent sheaves on Noetherian spaces
    • 56.13. Devissage of coherent sheaves
    • 56.14. Limits of coherent modules
    • 56.15. Vanishing cohomology
    • 56.16. Finite morphisms and affines
    • 56.17. A weak version of Chow's lemma
    • 56.18. Noetherian valuative criterion
    • 56.19. Higher direct images of coherent sheaves
    • 56.20. The theorem on formal functions
    • 56.21. Applications of the theorem on formal functions
    • 56.22. Other chapters
  • Chapter 57. Limits of Algebraic Spaces
    • 57.1. Introduction
    • 57.2. Conventions
    • 57.3. Morphisms of finite presentation
    • 57.4. Limits of algebraic spaces
    • 57.5. Descending properties
    • 57.6. Descending properties of morphisms
    • 57.7. Descending relative objects
    • 57.8. Absolute Noetherian approximation
    • 57.9. Applications
    • 57.10. Relative approximation
    • 57.11. Finite type closed in finite presentation
    • 57.12. Approximating proper morphisms
    • 57.13. Embedding into affine space
    • 57.14. Sections with support in a closed subset
    • 57.15. Characterizing affine spaces
    • 57.16. Finite cover by a scheme
    • 57.17. Obtaining schemes
    • 57.18. Application to modifications
    • 57.19. Other chapters
  • Chapter 58. Divisors on Algebraic Spaces
    • 58.1. Introduction
    • 58.2. Effective Cartier divisors
    • 58.3. Relative Proj
    • 58.4. Functoriality of relative proj
    • 58.5. Closed subspaces of relative proj
    • 58.6. Blowing up
    • 58.7. Strict transform
    • 58.8. Admissible blowups
    • 58.9. Other chapters
  • Chapter 59. Algebraic Spaces over Fields
    • 59.1. Introduction
    • 59.2. Conventions
    • 59.3. Generically finite morphisms
    • 59.4. Integral algebraic spaces
    • 59.5. Modifications and alterations
    • 59.6. Schematic locus
    • 59.7. Schematic locus and field extension
    • 59.8. Geometrically connected algebraic spaces
    • 59.9. Spaces smooth over fields
    • 59.10. Other chapters
  • Chapter 60. Topologies on Algebraic Spaces
    • 60.1. Introduction
    • 60.2. The general procedure
    • 60.3. Fpqc topology
    • 60.4. Fppf topology
    • 60.5. Syntomic topology
    • 60.6. Smooth topology
    • 60.7. Étale topology
    • 60.8. Zariski topology
    • 60.9. Other chapters
  • Chapter 61. Descent and Algebraic Spaces
    • 61.1. Introduction
    • 61.2. Conventions
    • 61.3. Descent data for quasi-coherent sheaves
    • 61.4. Fpqc descent of quasi-coherent sheaves
    • 61.5. Descent of finiteness properties of modules
    • 61.6. Fpqc coverings
    • 61.7. Descent of finiteness properties of morphisms
    • 61.8. Descending properties of spaces
    • 61.9. Descending properties of morphisms
    • 61.10. Descending properties of morphisms in the fpqc topology
    • 61.11. Descending properties of morphisms in the fppf topology
    • 61.12. Properties of morphisms local on the source
    • 61.13. Properties of morphisms local in the fpqc topology on the source
    • 61.14. Properties of morphisms local in the fppf topology on the source
    • 61.15. Properties of morphisms local in the syntomic topology on the source
    • 61.16. Properties of morphisms local in the smooth topology on the source
    • 61.17. Properties of morphisms local in the étale topology on the source
    • 61.18. Properties of morphisms smooth local on source-and-target
    • 61.19. Descent data for spaces over spaces
    • 61.20. Descent data in terms of sheaves
    • 61.21. Other chapters
  • Chapter 62. Derived Categories of Spaces
    • 62.1. Introduction
    • 62.2. Conventions
    • 62.3. Generalities
    • 62.4. Derived category of quasi-coherent modules on the small étale site
    • 62.5. Derived category of quasi-coherent modules
    • 62.6. Total direct image
    • 62.7. Derived category of coherent modules
    • 62.8. Induction principle
    • 62.9. Mayer-Vietoris
    • 62.10. The coherator
    • 62.11. The coherator for Noetherian spaces
    • 62.12. Pseudo-coherent and perfect complexes
    • 62.13. Approximation by perfect complexes
    • 62.14. Generating derived categories
    • 62.15. Compact and perfect objects
    • 62.16. Derived categories as module categories
    • 62.17. Cohomology and base change, IV
    • 62.18. Producing perfect complexes
    • 62.19. Computing Ext groups and base change
    • 62.20. Limits and derived categories
    • 62.21. Cohomology and base change, V
    • 62.22. Other chapters
  • Chapter 63. More on Morphisms of Spaces
    • 63.1. Introduction
    • 63.2. Conventions
    • 63.3. Radicial morphisms
    • 63.4. Monomorphisms
    • 63.5. Conormal sheaf of an immersion
    • 63.6. The normal cone of an immersion
    • 63.7. Sheaf of differentials of a morphism
    • 63.8. Topological invariance of the étale site
    • 63.9. Thickenings
    • 63.10. First order infinitesimal neighbourhood
    • 63.11. Formally smooth, étale, unramified transformations
    • 63.12. Formally unramified morphisms
    • 63.13. Universal first order thickenings
    • 63.14. Formally étale morphisms
    • 63.15. Infinitesimal deformations of maps
    • 63.16. Infinitesimal deformations of algebraic spaces
    • 63.17. Formally smooth morphisms
    • 63.18. Smoothness over a Noetherian base
    • 63.19. Openness of the flat locus
    • 63.20. Critère de platitude par fibres
    • 63.21. Flatness over a Noetherian base
    • 63.22. Normalization revisited
    • 63.23. Slicing Cohen-Macaulay morphisms
    • 63.24. Étale localization of morphisms
    • 63.25. Zariski's Main Theorem
    • 63.26. Stein factorization
    • 63.27. Extending properties from an open
    • 63.28. Blowing up and flatness
    • 63.29. Applications
    • 63.30. Chow's lemma
    • 63.31. Variants of Chow's Lemma
    • 63.32. Grothendieck's existence theorem
    • 63.33. Grothendieck's algebraization theorem
    • 63.34. Regular immersions
    • 63.35. Pseudo-coherent morphisms
    • 63.36. Perfect morphisms
    • 63.37. Local complete intersection morphisms
    • 63.38. When is a morphism an isomorphism?
    • 63.39. Exact sequences of differentials and conormal sheaves
    • 63.40. Other chapters
  • Chapter 64. Pushouts of Algebraic Spaces
    • 64.1. Introduction
    • 64.2. Pushouts in the category of algebraic spaces
    • 64.3. Formal glueing of quasi-coherent modules
    • 64.4. Formal glueing of algebraic spaces
    • 64.5. Coequalizers and glueing
    • 64.6. Other chapters
  • Chapter 65. Groupoids in Algebraic Spaces
    • 65.1. Introduction
    • 65.2. Conventions
    • 65.3. Notation
    • 65.4. Equivalence relations
    • 65.5. Group algebraic spaces
    • 65.6. Properties of group algebraic spaces
    • 65.7. Examples of group algebraic spaces
    • 65.8. Actions of group algebraic spaces
    • 65.9. Principal homogeneous spaces
    • 65.10. Equivariant quasi-coherent sheaves
    • 65.11. Groupoids in algebraic spaces
    • 65.12. Quasi-coherent sheaves on groupoids
    • 65.13. Crystals in quasi-coherent sheaves
    • 65.14. Groupoids and group spaces
    • 65.15. The stabilizer group algebraic space
    • 65.16. Restricting groupoids
    • 65.17. Invariant subspaces
    • 65.18. Quotient sheaves
    • 65.19. Quotient stacks
    • 65.20. Functoriality of quotient stacks
    • 65.21. The 2-cartesian square of a quotient stack
    • 65.22. The 2-coequalizer property of a quotient stack
    • 65.23. Explicit description of quotient stacks
    • 65.24. Restriction and quotient stacks
    • 65.25. Inertia and quotient stacks
    • 65.26. Gerbes and quotient stacks
    • 65.27. Quotient stacks and change of big site
    • 65.28. Separation conditions
    • 65.29. Other chapters
  • Chapter 66. More on Groupoids in Spaces
    • 66.1. Introduction
    • 66.2. Notation
    • 66.3. Useful diagrams
    • 66.4. Properties of groupoids
    • 66.5. Comparing fibres
    • 66.6. Restricting groupoids
    • 66.7. Properties of groups over fields and groupoids on fields
    • 66.8. Group algebraic spaces over fields
    • 66.9. No rational curves on groups
    • 66.10. The finite part of a morphism
    • 66.11. Finite collections of arrows
    • 66.12. The finite part of a groupoid
    • 66.13. Étale localization of groupoid schemes
    • 66.14. Other chapters
  • Chapter 67. Bootstrap
    • 67.1. Introduction
    • 67.2. Conventions
    • 67.3. Morphisms representable by algebraic spaces
    • 67.4. Properties of maps of presheaves representable by algebraic spaces
    • 67.5. Bootstrapping the diagonal
    • 67.6. Bootstrap
    • 67.7. Finding opens
    • 67.8. Slicing equivalence relations
    • 67.9. Quotient by a subgroupoid
    • 67.10. Final bootstrap
    • 67.11. Applications
    • 67.12. Algebraic spaces in the étale topology
    • 67.13. Other chapters
  • Chapter 68. Quotients of Groupoids
    • 68.1. Introduction
    • 68.2. Conventions and notation
    • 68.3. Invariant morphisms
    • 68.4. Categorical quotients
    • 68.5. Quotients as orbit spaces
    • 68.6. Coarse quotients
    • 68.7. Topological properties
    • 68.8. Invariant functions
    • 68.9. Good quotients
    • 68.10. Geometric quotients
    • 68.11. Other chapters
  • Chapter 69. Simplicial Spaces
    • 69.1. Introduction
    • 69.2. Simplicial topological spaces
    • 69.3. Simplicial sites and topoi
    • 69.4. Simplicial semi-representable objects
    • 69.5. Hypercovering in a site
    • 69.6. Proper hypercoverings in topology
    • 69.7. Simplicial schemes
    • 69.8. Descent in terms of simplicial schemes
    • 69.9. Quasi-coherent modules on simplicial schemes
    • 69.10. Groupoids and simplicial schemes
    • 69.11. Descent data give equivalence relations
    • 69.12. An example case
    • 69.13. Other chapters
  • Chapter 70. Formal Algebraic Spaces
    • 70.1. Introduction
    • 70.2. Formal schemes à la EGA
    • 70.3. Conventions and notation
    • 70.4. Topological rings and modules
    • 70.5. Affine formal algebraic spaces
    • 70.6. Countably indexed affine formal algebraic spaces
    • 70.7. Formal algebraic spaces
    • 70.8. Colimits of algebraic spaces along thickenings
    • 70.9. Completion along a closed subset
    • 70.10. Fibre products
    • 70.11. Separation axioms for formal algebraic spaces
    • 70.12. Quasi-compact formal algebraic spaces
    • 70.13. Quasi-compact and quasi-separated formal algebraic spaces
    • 70.14. Morphisms representable by algebraic spaces
    • 70.15. Types of formal algebraic spaces
    • 70.16. Morphisms and continuous ring maps
    • 70.17. Adic morphisms
    • 70.18. Morphisms of finite type
    • 70.19. Monomorphisms
    • 70.20. Closed immersions
    • 70.21. Separation axioms for morphisms
    • 70.22. Proper morphisms
    • 70.23. Formal algebraic spaces and fpqc coverings
    • 70.24. Maps out of affine formal schemes
    • 70.25. Other chapters
  • Chapter 71. Restricted Power Series
    • 71.1. Introduction
    • 71.2. Restricted power series
    • 71.3. Algebras topologically of finite type
    • 71.4. Two categories
    • 71.5. A naive cotangent complex
    • 71.6. Rig-étale homomorphisms
    • 71.7. Rig-étale morphisms
    • 71.8. Glueing rings along a principal ideal
    • 71.9. Glueing rings along an ideal
    • 71.10. In case the base ring is a G-ring
    • 71.11. Rig-surjective morphisms
    • 71.12. Algebraization
    • 71.13. Application to modifications
    • 71.14. Other chapters
  • Chapter 72. Resolution of Surfaces Revisited
    • 72.1. Introduction
    • 72.2. Modifications
    • 72.3. Strategy
    • 72.4. Dominating by quadratic transformations
    • 72.5. Dominating by normalized blowups
    • 72.6. Base change to the completion
    • 72.7. Implied properties
    • 72.8. Resolution
    • 72.9. Examples
    • 72.10. Other chapters
  • Chapter 73. Formal Deformation Theory
    • 73.1. Introduction
    • 73.2. Notation and Conventions
    • 73.3. The base category
    • 73.4. The completed base category
    • 73.5. Categories cofibered in groupoids
    • 73.6. Prorepresentable functors and predeformation categories
    • 73.7. Formal objects and completion categories
    • 73.8. Smooth morphisms
    • 73.9. Schlessinger's conditions
    • 73.10. Tangent spaces of functors
    • 73.11. Tangent spaces of predeformation categories
    • 73.12. Versal formal objects
    • 73.13. Minimal versal formal objects
    • 73.14. Miniversal formal objects and tangent spaces
    • 73.15. Rim-Schlessinger conditions and deformation categories
    • 73.16. Lifts of objects
    • 73.17. Schlessinger's theorem on prorepresentable functors
    • 73.18. Infinitesimal automorphisms
    • 73.19. Groupoids in functors on an arbitrary category
    • 73.20. Groupoids in functors on the base category
    • 73.21. Smooth groupoids in functors on the base category
    • 73.22. Deformation categories as quotients of groupoids in functors
    • 73.23. Presentations of categories cofibered in groupoids
    • 73.24. Presentations of deformation categories
    • 73.25. Remarks regarding minimality
    • 73.26. Change of residue field
    • 73.27. Other chapters
  • Chapter 74. Deformation Theory
    • 74.1. Introduction
    • 74.2. Deformations of rings and the naive cotangent complex
    • 74.3. Thickenings of ringed spaces
    • 74.4. Modules on first order thickenings of ringed spaces
    • 74.5. Infinitesimal deformations of modules on ringed spaces
    • 74.6. Application to flat modules on flat thickenings of ringed spaces
    • 74.7. Deformations of ringed spaces and the naive cotangent complex
    • 74.8. Thickenings of ringed topoi
    • 74.9. Modules on first order thickenings of ringed topoi
    • 74.10. Infinitesimal deformations of modules on ringed topi
    • 74.11. Application to flat modules on flat thickenings of ringed topoi
    • 74.12. Deformations of ringed topoi and the naive cotangent complex
    • 74.13. Other chapters
  • Chapter 75. The Cotangent Complex
    • 75.1. Introduction
    • 75.2. Advice for the reader
    • 75.3. The cotangent complex of a ring map
    • 75.4. Simplicial resolutions and derived lower shriek
    • 75.5. Constructing a resolution
    • 75.6. Functoriality
    • 75.7. The fundamental triangle
    • 75.8. Localization and étale ring maps
    • 75.9. Smooth ring maps
    • 75.10. Comparison with the naive cotangent complex
    • 75.11. A spectral sequence of Quillen
    • 75.12. Comparison with Lichtenbaum-Schlessinger
    • 75.13. The cotangent complex of a local complete intersection
    • 75.14. Tensor products and the cotangent complex
    • 75.15. Deformations of ring maps and the cotangent complex
    • 75.16. The Atiyah class of a module
    • 75.17. The cotangent complex
    • 75.18. The Atiyah class of a sheaf of modules
    • 75.19. The cotangent complex of a morphism of ringed spaces
    • 75.20. Deformations of ringed spaces and the cotangent complex
    • 75.21. The cotangent complex of a morphism of ringed topoi
    • 75.22. Deformations of ringed topoi and the cotangent complex
    • 75.23. The cotangent complex of a morphism of schemes
    • 75.24. The cotangent complex of a scheme over a ring
    • 75.25. The cotangent complex of a morphism of algebraic spaces
    • 75.26. The cotangent complex of an algebraic space over a ring
    • 75.27. Fibre products of algebraic spaces and the cotangent complex
    • 75.28. Other chapters
  • Chapter 76. Algebraic Stacks
    • 76.1. Introduction
    • 76.2. Conventions
    • 76.3. Notation
    • 76.4. Representable categories fibred in groupoids
    • 76.5. The 2-Yoneda lemma
    • 76.6. Representable morphisms of categories fibred in groupoids
    • 76.7. Split categories fibred in groupoids
    • 76.8. Categories fibred in groupoids representable by algebraic spaces
    • 76.9. Morphisms representable by algebraic spaces
    • 76.10. Properties of morphisms representable by algebraic spaces
    • 76.11. Stacks in groupoids
    • 76.12. Algebraic stacks
    • 76.13. Algebraic stacks and algebraic spaces
    • 76.14. 2-Fibre products of algebraic stacks
    • 76.15. Algebraic stacks, overhauled
    • 76.16. From an algebraic stack to a presentation
    • 76.17. The algebraic stack associated to a smooth groupoid
    • 76.18. Change of big site
    • 76.19. Change of base scheme
    • 76.20. Other chapters
  • Chapter 77. Examples of Stacks
    • 77.1. Introduction
    • 77.2. Notation
    • 77.3. Examples of stacks
    • 77.4. Quasi-coherent sheaves
    • 77.5. The stack of finitely generated quasi-coherent sheaves
    • 77.6. Finite étale covers
    • 77.7. Algebraic spaces
    • 77.8. The stack of finite type algebraic spaces
    • 77.9. Examples of stacks in groupoids
    • 77.10. The stack associated to a sheaf
    • 77.11. The stack in groupoids of finitely generated quasi-coherent sheaves
    • 77.12. The stack in groupoids of finite type algebraic spaces
    • 77.13. Quotient stacks
    • 77.14. Classifying torsors
    • 77.15. Quotients by group actions
    • 77.16. The Picard stack
    • 77.17. Examples of inertia stacks
    • 77.18. Finite Hilbert stacks
    • 77.19. Other chapters
  • Chapter 78. Sheaves on Algebraic Stacks
    • 78.1. Introduction
    • 78.2. Conventions
    • 78.3. Presheaves
    • 78.4. Sheaves
    • 78.5. Computing pushforward
    • 78.6. The structure sheaf
    • 78.7. Sheaves of modules
    • 78.8. Representable categories
    • 78.9. Restriction
    • 78.10. Restriction to algebraic spaces
    • 78.11. Quasi-coherent modules
    • 78.12. Stackification and sheaves
    • 78.13. Quasi-coherent sheaves and presentations
    • 78.14. Quasi-coherent sheaves on algebraic stacks
    • 78.15. Cohomology
    • 78.16. Injective sheaves
    • 78.17. The Cech complex
    • 78.18. The relative Cech complex
    • 78.19. Cohomology on algebraic stacks
    • 78.20. Higher direct images and algebraic stacks
    • 78.21. Comparison
    • 78.22. Change of topology
    • 78.23. Other chapters
  • Chapter 79. Criteria for Representability
    • 79.1. Introduction
    • 79.2. Conventions
    • 79.3. What we already know
    • 79.4. Morphisms of stacks in groupoids
    • 79.5. Limit preserving on objects
    • 79.6. Formally smooth on objects
    • 79.7. Surjective on objects
    • 79.8. Algebraic morphisms
    • 79.9. Spaces of sections
    • 79.10. Relative morphisms
    • 79.11. Restriction of scalars
    • 79.12. Finite Hilbert stacks
    • 79.13. The finite Hilbert stack of a point
    • 79.14. Finite Hilbert stacks of spaces
    • 79.15. LCI locus in the Hilbert stack
    • 79.16. Bootstrapping algebraic stacks
    • 79.17. Applications
    • 79.18. When is a quotient stack algebraic?
    • 79.19. Algebraic stacks in the étale topology
    • 79.20. Other chapters
  • Chapter 80. Artin's axioms
    • 80.1. Introduction
    • 80.2. Conventions
    • 80.3. Predeformation categories
    • 80.4. Pushouts and stacks
    • 80.5. The Rim-Schlessinger condition
    • 80.6. Deformation categories
    • 80.7. Change of field
    • 80.8. Tangent spaces
    • 80.9. Formal objects
    • 80.10. Approximation
    • 80.11. Versality
    • 80.12. Axioms
    • 80.13. Limit preserving
    • 80.14. Openness of versality
    • 80.15. Axioms for functors
    • 80.16. Algebraic spaces
    • 80.17. Algebraic stacks
    • 80.18. Infinitesimal deformations
    • 80.19. Obstruction theories
    • 80.20. Naive obstruction theories
    • 80.21. A dual notion
    • 80.22. Examples of deformation problems
    • 80.23. Other chapters
  • Chapter 81. Quot and Hilbert Spaces
    • 81.1. Introduction
    • 81.2. Conventions
    • 81.3. The Hom functor
    • 81.4. The Isom functor
    • 81.5. The stack of coherent sheaves
    • 81.6. The stack of coherent sheaves in the non-flat case
    • 81.7. Flattening functors
    • 81.8. The functor of quotients
    • 81.9. The quot functor
    • 81.10. Other chapters
  • Chapter 82. Properties of Algebraic Stacks
    • 82.1. Introduction
    • 82.2. Conventions and abuse of language
    • 82.3. Properties of morphisms representable by algebraic spaces
    • 82.4. Points of algebraic stacks
    • 82.5. Surjective morphisms
    • 82.6. Quasi-compact algebraic stacks
    • 82.7. Properties of algebraic stacks defined by properties of schemes
    • 82.8. Monomorphisms of algebraic stacks
    • 82.9. Immersions of algebraic stacks
    • 82.10. Reduced algebraic stacks
    • 82.11. Residual gerbes
    • 82.12. Dimension of a stack
    • 82.13. Other chapters
  • Chapter 83. Morphisms of Algebraic Stacks
    • 83.1. Introduction
    • 83.2. Conventions and abuse of language
    • 83.3. Properties of diagonals
    • 83.4. Separation axioms
    • 83.5. Inertia stacks
    • 83.6. Higher diagonals
    • 83.7. Quasi-compact morphisms
    • 83.8. Noetherian algebraic stacks
    • 83.9. Open morphisms
    • 83.10. Submersive morphisms
    • 83.11. Universally closed morphisms
    • 83.12. Types of morphisms smooth local on source-and-target
    • 83.13. Morphisms of finite type
    • 83.14. Points of finite type
    • 83.15. Special presentations of algebraic stacks
    • 83.16. Quasi-finite morphisms
    • 83.17. Flat morphisms
    • 83.18. Morphisms of finite presentation
    • 83.19. Gerbes
    • 83.20. Stratification by gerbes
    • 83.21. Existence of residual gerbes
    • 83.22. Smooth morphisms
    • 83.23. Other chapters
  • Chapter 84. Cohomology of Algebraic Stacks
    • 84.1. Introduction
    • 84.2. Conventions and abuse of language
    • 84.3. Notation
    • 84.4. Pullback of quasi-coherent modules
    • 84.5. The key lemma
    • 84.6. Locally quasi-coherent modules
    • 84.7. Flat comparison maps
    • 84.8. Parasitic modules
    • 84.9. Quasi-coherent modules, I
    • 84.10. Pushforward of quasi-coherent modules
    • 84.11. The lisse-étale and the flat-fppf sites
    • 84.12. Quasi-coherent modules, II
    • 84.13. Other chapters
  • Chapter 85. Derived Categories of Stacks
    • 85.1. Introduction
    • 85.2. Conventions, notation, and abuse of language
    • 85.3. The lisse-étale and the flat-fppf sites
    • 85.4. Derived categories of quasi-coherent modules
    • 85.5. Derived pushforward of quasi-coherent modules
    • 85.6. Derived pullback of quasi-coherent modules
    • 85.7. Other chapters
  • Chapter 86. Introducing Algebraic Stacks
    • 86.1. Why read this?
    • 86.2. Preliminary
    • 86.3. The moduli stack of elliptic curves
    • 86.4. Fibre products
    • 86.5. The definition
    • 86.6. A smooth cover
    • 86.7. Properties of algebraic stacks
    • 86.8. Other chapters
  • Chapter 87. More on Morphisms of Stacks
    • 87.1. Introduction
    • 87.2. Conventions and abuse of language
    • 87.3. Thickenings
    • 87.4. Other chapters
  • Chapter 88. Examples
    • 88.1. Introduction
    • 88.2. An empty limit
    • 88.3. A zero limit
    • 88.4. Non-quasi-compact inverse limit of quasi-compact spaces
    • 88.5. A nonintegral connected scheme whose local rings are domains
    • 88.6. Noncomplete completion
    • 88.7. Noncomplete quotient
    • 88.8. Completion is not exact
    • 88.9. The category of complete modules is not abelian
    • 88.10. The category of derived complete modules
    • 88.11. Nonflat completions
    • 88.12. Nonabelian category of quasi-coherent modules
    • 88.13. Regular sequences and base change
    • 88.14. A Noetherian ring of infinite dimension
    • 88.15. Local rings with nonreduced completion
    • 88.16. A non catenary Noetherian local ring
    • 88.17. Existence of bad local Noetherian rings
    • 88.18. Non-quasi-affine variety with quasi-affine normalization
    • 88.19. A locally closed subscheme which is not open in closed
    • 88.20. Nonexistence of suitable opens
    • 88.21. Nonexistence of quasi-compact dense open subscheme
    • 88.22. Affines over algebraic spaces
    • 88.23. Pushforward of quasi-coherent modules
    • 88.24. A nonfinite module with finite free rank 1 stalks
    • 88.25. A finite flat module which is not projective
    • 88.26. A projective module which is not locally free
    • 88.27. Zero dimensional local ring with nonzero flat ideal
    • 88.28. An epimorphism of zero-dimensional rings which is not surjective
    • 88.29. Finite type, not finitely presented, flat at prime
    • 88.30. Finite type, flat and not of finite presentation
    • 88.31. Topology of a finite type ring map
    • 88.32. Pure not universally pure
    • 88.33. A formally smooth non-flat ring map
    • 88.34. A formally étale non-flat ring map
    • 88.35. A formally étale ring map with nontrivial cotangent complex
    • 88.36. Ideals generated by sets of idempotents and localization
    • 88.37. A ring map which identifies local rings which is not ind-étale
    • 88.38. Non flasque quasi-coherent sheaf associated to injective module
    • 88.39. A non-separated flat group scheme
    • 88.40. A non-flat group scheme with flat identity component
    • 88.41. A non-separated group algebraic space over a field
    • 88.42. Specializations between points in fibre étale morphism
    • 88.43. A torsor which is not an fppf torsor
    • 88.44. Stack with quasi-compact flat covering which is not algebraic
    • 88.45. Limit preserving on objects, not limit preserving
    • 88.46. A non-algebraic classifying stack
    • 88.47. Sheaf with quasi-compact flat covering which is not algebraic
    • 88.48. Sheaves and specializations
    • 88.49. Sheaves and constructible functions
    • 88.50. The lisse-étale site is not functorial
    • 88.51. Derived pushforward of quasi-coherent modules
    • 88.52. A big abelian category
    • 88.53. Weakly associated points and scheme theoretic density
    • 88.54. Example of non-additivity of traces
    • 88.55. Being projective is not local on the base
    • 88.56. Descent data for schemes need not be effective, even for a projective morphism
    • 88.57. Derived base change
    • 88.58. An interesting compact object
    • 88.59. Two differential graded categories
    • 88.60. An example of a non-algebraic Hom-stack
    • 88.61. A counter example to Grothendieck's existence theorem
    • 88.62. Affine formal algebraic spaces
    • 88.63. Flat maps are not directed limits of finitely presented flat maps
    • 88.64. The category of modules modulo torsion modules
    • 88.65. Different colimit topologies
    • 88.66. Other chapters
  • Chapter 89. Exercises
    • 89.1. Algebra
    • 89.2. Colimits
    • 89.3. Additive and abelian categories
    • 89.4. Flat ring maps
    • 89.5. The Spectrum of a ring
    • 89.6. Localization
    • 89.7. Nakayama's Lemma
    • 89.8. Length
    • 89.9. Singularities
    • 89.10. Hilbert Nullstellensatz
    • 89.11. Dimension
    • 89.12. Catenary rings
    • 89.13. Fraction fields
    • 89.14. Transcendence degree
    • 89.15. Finite locally free modules
    • 89.16. Glueing
    • 89.17. Going up and going down
    • 89.18. Fitting ideals
    • 89.19. Hilbert functions
    • 89.20. Proj of a ring
    • 89.21. Cohen-Macaulay rings of dimension 1
    • 89.22. Infinitely many primes
    • 89.23. Filtered derived category
    • 89.24. Regular functions
    • 89.25. Sheaves
    • 89.26. Schemes
    • 89.27. Morphisms
    • 89.28. Tangent Spaces
    • 89.29. Quasi-coherent Sheaves
    • 89.30. Proj and projective schemes
    • 89.31. Morphisms from surfaces to curves
    • 89.32. Invertible sheaves
    • 89.33. Cech Cohomology
    • 89.34. Divisors
    • 89.35. Differentials
    • 89.36. Schemes, Final Exam, Fall 2007
    • 89.37. Schemes, Final Exam, Spring 2009
    • 89.38. Schemes, Final Exam, Fall 2010
    • 89.39. Schemes, Final Exam, Spring 2011
    • 89.40. Schemes, Final Exam, Fall 2011
    • 89.41. Schemes, Final Exam, Fall 2013
    • 89.42. Schemes, Final Exam, Spring 2014
    • 89.43. Other chapters
  • Chapter 90. A Guide to the Literature
    • 90.1. Short introductory articles
    • 90.2. Classic references
    • 90.3. Books and online notes
    • 90.4. Related references on foundations of stacks
    • 90.5. Papers in the literature
    • 90.6. Stacks in other fields
    • 90.7. Higher stacks
    • 90.8. Other chapters
  • Chapter 91. Desirables
    • 91.1. Introduction
    • 91.2. Conventions
    • 91.3. Sites and Topoi
    • 91.4. Stacks
    • 91.5. Simplicial methods
    • 91.6. Cohomology of schemes
    • 91.7. Deformation theory à la Schlessinger
    • 91.8. Definition of algebraic stacks
    • 91.9. Examples of schemes, algebraic spaces, algebraic stacks
    • 91.10. Properties of algebraic stacks
    • 91.11. Lisse étale site of an algebraic stack
    • 91.12. Things you always wanted to know but were afraid to ask
    • 91.13. Quasi-coherent sheaves on stacks
    • 91.14. Flat and smooth
    • 91.15. Artin's representability theorem
    • 91.16. DM stacks are finitely covered by schemes
    • 91.17. Martin Olsson's paper on properness
    • 91.18. Proper pushforward of coherent sheaves
    • 91.19. Keel and Mori
    • 91.20. Add more here
    • 91.21. Other chapters
  • Chapter 92. Coding Style
    • 92.1. List of style comments
    • 92.2. Other chapters
  • Chapter 93. Obsolete
    • 93.1. Introduction
    • 93.2. Homological algebra
    • 93.3. Obsolete algebra lemmas
    • 93.4. Lemmas related to ZMT
    • 93.5. Formally smooth ring maps
    • 93.6. Cohomology
    • 93.7. Simplicial methods
    • 93.8. Obsolete lemmas on schemes
    • 93.9. Functor of quotients
    • 93.10. Spaces and fpqc coverings
    • 93.11. Very reasonable algebraic spaces
    • 93.12. Variants of cotangent complexes for schemes
    • 93.13. Deformations and obstructions of flat modules
    • 93.14. Modifications
    • 93.15. Intersection theory
    • 93.16. Duplicate references
    • 93.17. Other chapters
  • Chapter 94. GNU Free Documentation License
    • 94.1. APPLICABILITY AND DEFINITIONS
    • 94.2. VERBATIM COPYING
    • 94.3. COPYING IN QUANTITY
    • 94.4. MODIFICATIONS
    • 94.5. COMBINING DOCUMENTS
    • 94.6. COLLECTIONS OF DOCUMENTS
    • 94.7. AGGREGATION WITH INDEPENDENT WORKS
    • 94.8. TRANSLATION
    • 94.9. TERMINATION
    • 94.10. FUTURE REVISIONS OF THIS LICENSE
    • 94.11. ADDENDUM: How to use this License for your documents
    • 94.12. Other chapters
  • Chapter 95. Auto generated index
    • 95.1. Alphabetized definitions
    • 95.2. Definitions listed per chapter
    • 95.3. Other chapters
  • Bibliography
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