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Contents
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Language
English
ISBN
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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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