2 Schemes

• Chapter 26: Schemes
  • Section 26.1: Introduction
  • Section 26.2: Locally ringed spaces
  • Section 26.3: Open immersions of locally ringed spaces
  • Section 26.4: Closed immersions of locally ringed spaces
  • Section 26.5: Affine schemes
  • Section 26.6: The category of affine schemes
  • Section 26.7: Quasi-coherent sheaves on affines
  • Section 26.8: Closed subspaces of affine schemes
  • Section 26.9: Schemes
  • Section 26.10: Immersions of schemes
  • Section 26.11: Zariski topology of schemes
  • Section 26.12: Reduced schemes
  • Section 26.13: Points of schemes
  • Section 26.14: Glueing schemes
  • Section 26.15: A representability criterion
  • Section 26.16: Existence of fibre products of schemes
  • Section 26.17: Fibre products of schemes
  • Section 26.18: Base change in algebraic geometry
  • Section 26.19: Quasi-compact morphisms
  • Section 26.20: Valuative criterion for universal closedness
  • Section 26.21: Separation axioms
  • Section 26.22: Valuative criterion of separatedness
  • Section 26.23: Monomorphisms
  • Section 26.24: Functoriality for quasi-coherent modules
• Chapter 27: Constructions of Schemes
  • Section 27.1: Introduction
  • Section 27.2: Relative glueing
  • Section 27.3: Relative spectrum via glueing
  • Section 27.4: Relative spectrum as a functor
  • Section 27.5: Affine n-space
  • Section 27.6: Vector bundles
  • Section 27.7: Cones
  • Section 27.8: Proj of a graded ring
  • Section 27.9: Quasi-coherent sheaves on Proj
  • Section 27.10: Invertible sheaves on Proj
  • Section 27.11: Functoriality of Proj
  • Section 27.12: Morphisms into Proj
  • Section 27.13: Projective space
  • Section 27.14: Invertible sheaves and morphisms into Proj
  • Section 27.15: Relative Proj via glueing
  • Section 27.16: Relative Proj as a functor
  • Section 27.17: Quasi-coherent sheaves on relative Proj
  • Section 27.18: Functoriality of relative Proj
  • Section 27.19: Invertible sheaves and morphisms into relative Proj
  • Section 27.20: Twisting by invertible sheaves and relative Proj
  • Section 27.21: Projective bundles
  • Section 27.22: Grassmannians
• Chapter 28: Properties of Schemes
  • Section 28.1: Introduction
  • Section 28.2: Constructible sets
  • Section 28.3: Integral, irreducible, and reduced schemes
  • Section 28.4: Types of schemes defined by properties of rings
  • Section 28.5: Noetherian schemes
  • Section 28.6: Jacobson schemes
  • Section 28.7: Normal schemes
  • Section 28.8: Cohen-Macaulay schemes
  • Section 28.9: Regular schemes
  • Section 28.10: Dimension
  • Section 28.11: Catenary schemes
  • Section 28.12: Serre's conditions
  • Section 28.13: Japanese and Nagata schemes
  • Section 28.14: The singular locus
  • Section 28.15: Local irreducibility
  • Section 28.16: Characterizing modules of finite type and finite presentation
  • Section 28.17: Sections over principal opens
  • Section 28.18: Quasi-affine schemes
  • Section 28.19: Flat modules
  • Section 28.20: Locally free modules
  • Section 28.21: Locally projective modules
  • Section 28.22: Extending quasi-coherent sheaves
  • Section 28.23: Gabber's result
  • Section 28.24: Sections with support in a closed subset
  • Section 28.25: Sections of quasi-coherent sheaves
  • Section 28.26: Ample invertible sheaves
  • Section 28.27: Affine and quasi-affine schemes
  • Section 28.28: Quasi-coherent sheaves and ample invertible sheaves
  • Section 28.29: Finding suitable affine opens
• Chapter 29: Morphisms of Schemes
  • Section 29.1: Introduction
  • Section 29.2: Closed immersions
  • Section 29.3: Immersions
  • Section 29.4: Closed immersions and quasi-coherent sheaves
  • Section 29.5: Supports of modules
  • Section 29.6: Scheme theoretic image
  • Section 29.7: Scheme theoretic closure and density
  • Section 29.8: Dominant morphisms
  • Section 29.9: Surjective morphisms
  • Section 29.10: Radicial and universally injective morphisms
  • Section 29.11: Affine morphisms
  • Section 29.12: Families of ample invertible modules
  • Section 29.13: Quasi-affine morphisms
  • Section 29.14: Types of morphisms defined by properties of ring maps
  • Section 29.15: Morphisms of finite type
  • Section 29.16: Points of finite type and Jacobson schemes
  • Section 29.17: Universally catenary schemes
  • Section 29.18: Nagata schemes, reprise
  • Section 29.19: The singular locus, reprise
  • Section 29.20: Quasi-finite morphisms
  • Section 29.21: Morphisms of finite presentation
  • Section 29.22: Constructible sets
  • Section 29.23: Open morphisms
  • Section 29.24: Submersive morphisms
  • Section 29.25: Flat morphisms
  • Section 29.26: Flat closed immersions
  • Section 29.27: Generic flatness
  • Section 29.28: Morphisms and dimensions of fibres
  • Section 29.29: Morphisms of given relative dimension
  • Section 29.30: Syntomic morphisms
  • Section 29.31: Conormal sheaf of an immersion
  • Section 29.32: Sheaf of differentials of a morphism
  • Section 29.33: Finite order differential operators
  • Section 29.34: Smooth morphisms
  • Section 29.35: Unramified morphisms
  • Section 29.36: Étale morphisms
  • Section 29.37: Relatively ample sheaves
  • Section 29.38: Very ample sheaves
  • Section 29.39: Ample and very ample sheaves relative to finite type morphisms
  • Section 29.40: Quasi-projective morphisms
  • Section 29.41: Proper morphisms
  • Section 29.42: Valuative criteria
  • Section 29.43: Projective morphisms
  • Section 29.44: Integral and finite morphisms
  • Section 29.45: Universal homeomorphisms
  • Section 29.46: Universal homeomorphisms of affine schemes
  • Section 29.47: Absolute weak normalization and seminormalization
  • Section 29.48: Finite locally free morphisms
  • Section 29.49: Rational maps
  • Section 29.50: Birational morphisms
  • Section 29.51: Generically finite morphisms
  • Section 29.52: The dimension formula
  • Section 29.53: Relative normalization
  • Section 29.54: Normalization
  • Section 29.55: Weak normalization
  • Section 29.56: Zariski's Main Theorem (algebraic version)
  • Section 29.57: Universally bounded fibres
  • Section 29.58: Miscellany
• Chapter 30: Cohomology of Schemes
  • Section 30.1: Introduction
  • Section 30.2: Čech cohomology of quasi-coherent sheaves
  • Section 30.3: Vanishing of cohomology
  • Section 30.4: Quasi-coherence of higher direct images
  • Section 30.5: Cohomology and base change, I
  • Section 30.6: Colimits and higher direct images
  • Section 30.7: Cohomology and base change, II
  • Section 30.8: Cohomology of projective space
  • Section 30.9: Coherent sheaves on locally Noetherian schemes
  • Section 30.10: Coherent sheaves on Noetherian schemes
  • Section 30.11: Depth
  • Section 30.12: Devissage of coherent sheaves
  • Section 30.13: Finite morphisms and affines
  • Section 30.14: Coherent sheaves on Proj, I
  • Section 30.15: Coherent sheaves on Proj, II
  • Section 30.16: Higher direct images along projective morphisms
  • Section 30.17: Ample invertible sheaves and cohomology
  • Section 30.18: Chow's Lemma
  • Section 30.19: Higher direct images of coherent sheaves
  • Section 30.20: The theorem on formal functions
  • Section 30.21: Applications of the theorem on formal functions
  • Section 30.22: Cohomology and base change, III
  • Section 30.23: Coherent formal modules
  • Section 30.24: Grothendieck's existence theorem, I
  • Section 30.25: Grothendieck's existence theorem, II
  • Section 30.26: Being proper over a base
  • Section 30.27: Grothendieck's existence theorem, III
  • Section 30.28: Grothendieck's algebraization theorem
• Chapter 31: Divisors
  • Section 31.1: Introduction
  • Section 31.2: Associated points
  • Section 31.3: Morphisms and associated points
  • Section 31.4: Embedded points
  • Section 31.5: Weakly associated points
  • Section 31.6: Morphisms and weakly associated points
  • Section 31.7: Relative assassin
  • Section 31.8: Relative weak assassin
  • Section 31.9: Fitting ideals
  • Section 31.10: The singular locus of a morphism
  • Section 31.11: Torsion free modules
  • Section 31.12: Reflexive modules
  • Section 31.13: Effective Cartier divisors
  • Section 31.14: Effective Cartier divisors and invertible sheaves
  • Section 31.15: Effective Cartier divisors on Noetherian schemes
  • Section 31.16: Complements of affine opens
  • Section 31.17: Norms
  • Section 31.18: Relative effective Cartier divisors
  • Section 31.19: The normal cone of an immersion
  • Section 31.20: Regular ideal sheaves
  • Section 31.21: Regular immersions
  • Section 31.22: Relative regular immersions
  • Section 31.23: Meromorphic functions and sections
  • Section 31.24: Meromorphic functions and sections; Noetherian case
  • Section 31.25: Meromorphic functions and sections; reduced case
  • Section 31.26: Weil divisors
  • Section 31.27: The Weil divisor class associated to an invertible module
  • Section 31.28: More on invertible modules
  • Section 31.29: Weil divisors on normal schemes
  • Section 31.30: Relative Proj
  • Section 31.31: Closed subschemes of relative proj
  • Section 31.32: Blowing up
  • Section 31.33: Strict transform
  • Section 31.34: Admissible blowups
  • Section 31.35: Blowing up and flatness
  • Section 31.36: Modifications
• Chapter 32: Limits of Schemes
  • Section 32.1: Introduction
  • Section 32.2: Directed limits of schemes with affine transition maps
  • Section 32.3: Infinite products
  • Section 32.4: Descending properties
  • Section 32.5: Absolute Noetherian Approximation
  • Section 32.6: Limits and morphisms of finite presentation
  • Section 32.7: Relative approximation
  • Section 32.8: Descending properties of morphisms
  • Section 32.9: Finite type closed in finite presentation
  • Section 32.10: Descending relative objects
  • Section 32.11: Characterizing affine schemes
  • Section 32.12: Variants of Chow's Lemma
  • Section 32.13: Applications of Chow's lemma
  • Section 32.14: Universally closed morphisms
  • Section 32.15: Noetherian valuative criterion
  • Section 32.16: Refined Noetherian valuative criteria
  • Section 32.17: Valuative criteria over a Nagata base
  • Section 32.18: Limits and dimensions of fibres
  • Section 32.19: Base change in top degree
  • Section 32.20: Glueing in closed fibres
  • Section 32.21: Application to modifications
  • Section 32.22: Descending finite type schemes
• Chapter 33: Varieties
  • Section 33.1: Introduction
  • Section 33.2: Notation
  • Section 33.3: Varieties
  • Section 33.4: Varieties and rational maps
  • Section 33.5: Change of fields and local rings
  • Section 33.6: Geometrically reduced schemes
  • Section 33.7: Geometrically connected schemes
  • Section 33.8: Geometrically irreducible schemes
  • Section 33.9: Geometrically integral schemes
  • Section 33.10: Geometrically normal schemes
  • Section 33.11: Change of fields and locally Noetherian schemes
  • Section 33.12: Geometrically regular schemes
  • Section 33.13: Change of fields and the Cohen-Macaulay property
  • Section 33.14: Change of fields and the Jacobson property
  • Section 33.15: Change of fields and ample invertible sheaves
  • Section 33.16: Tangent spaces
  • Section 33.17: Generically finite morphisms
  • Section 33.18: Variants of Noether normalization
  • Section 33.19: Dimension of fibres
  • Section 33.20: Algebraic schemes
  • Section 33.21: Complete local rings
  • Section 33.22: Global generation
  • Section 33.23: Separating points and tangent vectors
  • Section 33.24: Closures of products
  • Section 33.25: Schemes smooth over fields
  • Section 33.26: Types of varieties
  • Section 33.27: Normalization
  • Section 33.28: Groups of invertible functions
  • Section 33.29: Künneth formula, I
  • Section 33.30: Picard groups of varieties
  • Section 33.31: Uniqueness of base field
  • Section 33.32: Automorphisms
  • Section 33.33: Euler characteristics
  • Section 33.34: Projective space
  • Section 33.35: Coherent sheaves on projective space
  • Section 33.36: Frobenii
  • Section 33.37: Glueing dimension one rings
  • Section 33.38: One dimensional Noetherian schemes
  • Section 33.39: The delta invariant
  • Section 33.40: The number of branches
  • Section 33.41: Normalization of one dimensional schemes
  • Section 33.42: Finding affine opens
  • Section 33.43: Curves
  • Section 33.44: Degrees on curves
  • Section 33.45: Numerical intersections
  • Section 33.46: Embedding dimension
  • Section 33.47: Bertini theorems
  • Section 33.48: Enriques-Severi-Zariski
• Chapter 34: Topologies on Schemes
  • Section 34.1: Introduction
  • Section 34.2: The general procedure
  • Section 34.3: The Zariski topology
  • Section 34.4: The étale topology
  • Section 34.5: The smooth topology
  • Section 34.6: The syntomic topology
  • Section 34.7: The fppf topology
  • Section 34.8: The ph topology
  • Section 34.9: The fpqc topology
  • Section 34.10: The V topology
  • Section 34.11: Change of topologies
  • Section 34.12: Change of big sites
  • Section 34.13: Extending functors
• Chapter 35: Descent
  • Section 35.1: Introduction
  • Section 35.2: Descent data for quasi-coherent sheaves
  • Section 35.3: Descent for modules
  • Section 35.4: Descent for universally injective morphisms
  • Section 35.5: Fpqc descent of quasi-coherent sheaves
  • Section 35.6: Galois descent for quasi-coherent sheaves
  • Section 35.7: Descent of finiteness properties of modules
  • Section 35.8: Quasi-coherent sheaves and topologies, I
  • Section 35.9: Cohomology of quasi-coherent modules and topologies
  • Section 35.10: Quasi-coherent sheaves and topologies, II
  • Section 35.11: Quasi-coherent modules and affines
  • Section 35.12: Parasitic modules
  • Section 35.13: Fpqc coverings are universal effective epimorphisms
  • Section 35.14: Descent of finiteness and smoothness properties of morphisms
  • Section 35.15: Local properties of schemes
  • Section 35.16: Properties of schemes local in the fppf topology
  • Section 35.17: Properties of schemes local in the syntomic topology
  • Section 35.18: Properties of schemes local in the smooth topology
  • Section 35.19: Variants on descending properties
  • Section 35.20: Germs of schemes
  • Section 35.21: Local properties of germs
  • Section 35.22: Properties of morphisms local on the target
  • Section 35.23: Properties of morphisms local in the fpqc topology on the target
  • Section 35.24: Properties of morphisms local in the fppf topology on the target
  • Section 35.25: Application of fpqc descent of properties of morphisms
  • Section 35.26: Properties of morphisms local on the source
  • Section 35.27: Properties of morphisms local in the fpqc topology on the source
  • Section 35.28: Properties of morphisms local in the fppf topology on the source
  • Section 35.29: Properties of morphisms local in the syntomic topology on the source
  • Section 35.30: Properties of morphisms local in the smooth topology on the source
  • Section 35.31: Properties of morphisms local in the étale topology on the source
  • Section 35.32: Properties of morphisms étale local on source-and-target
  • Section 35.33: Properties of morphisms of germs local on source-and-target
  • Section 35.34: Descent data for schemes over schemes
  • Section 35.35: Fully faithfulness of the pullback functors
  • Section 35.36: Descending types of morphisms
  • Section 35.37: Descending affine morphisms
  • Section 35.38: Descending quasi-affine morphisms
  • Section 35.39: Descent data in terms of sheaves
• Chapter 36: Derived Categories of Schemes
  • Section 36.1: Introduction
  • Section 36.2: Conventions
  • Section 36.3: Derived category of quasi-coherent modules
  • Section 36.4: Total direct image
  • Section 36.5: Affine morphisms
  • Section 36.6: Cohomology with support in a closed subset
  • Section 36.7: The coherator
  • Section 36.8: The coherator for Noetherian schemes
  • Section 36.9: Koszul complexes
  • Section 36.10: Pseudo-coherent and perfect complexes
  • Section 36.11: Derived category of coherent modules
  • Section 36.12: Descent finiteness properties of complexes
  • Section 36.13: Lifting complexes
  • Section 36.14: Approximation by perfect complexes
  • Section 36.15: Generating derived categories
  • Section 36.16: An example generator
  • Section 36.17: Compact and perfect objects
  • Section 36.18: Derived categories as module categories
  • Section 36.19: Characterizing pseudo-coherent complexes, I
  • Section 36.20: An example equivalence
  • Section 36.21: The coherator revisited
  • Section 36.22: Cohomology and base change, IV
  • Section 36.23: Künneth formula, II
  • Section 36.24: Künneth formula, III
  • Section 36.25: Künneth formula for Ext
  • Section 36.26: Cohomology and base change, V
  • Section 36.27: Producing perfect complexes
  • Section 36.28: A projection formula for Ext
  • Section 36.29: Limits and derived categories
  • Section 36.30: Cohomology and base change, VI
  • Section 36.31: Perfect complexes
  • Section 36.32: Applications
  • Section 36.33: Other applications
  • Section 36.34: Characterizing pseudo-coherent complexes, II
  • Section 36.35: Relatively perfect objects
  • Section 36.36: The resolution property
  • Section 36.37: The resolution property and perfect complexes
  • Section 36.38: K-groups
  • Section 36.39: Determinants of complexes
  • Section 36.40: Detecting Boundedness
  • Section 36.41: Quasi-coherent objects in the derived category
• Chapter 37: More on Morphisms
  • Section 37.1: Introduction
  • Section 37.2: Thickenings
  • Section 37.3: Morphisms of thickenings
  • Section 37.4: Picard groups of thickenings
  • Section 37.5: First order infinitesimal neighbourhood
  • Section 37.6: Formally unramified morphisms
  • Section 37.7: Universal first order thickenings
  • Section 37.8: Formally étale morphisms
  • Section 37.9: Infinitesimal deformations of maps
  • Section 37.10: Infinitesimal deformations of schemes
  • Section 37.11: Formally smooth morphisms
  • Section 37.12: Smoothness over a Noetherian base
  • Section 37.13: The naive cotangent complex
  • Section 37.14: Pushouts in the category of schemes, I
  • Section 37.15: Openness of the flat locus
  • Section 37.16: Critère de platitude par fibres
  • Section 37.17: Closed immersions between smooth schemes
  • Section 37.18: Flat modules and relative assassins
  • Section 37.19: Normalization revisited
  • Section 37.20: Normal morphisms
  • Section 37.21: Regular morphisms
  • Section 37.22: Cohen-Macaulay morphisms
  • Section 37.23: Slicing Cohen-Macaulay morphisms
  • Section 37.24: Generic fibres
  • Section 37.25: Relative assassins
  • Section 37.26: Reduced fibres
  • Section 37.27: Irreducible components of fibres
  • Section 37.28: Connected components of fibres
  • Section 37.29: Connected components meeting a section
  • Section 37.30: Dimension of fibres
  • Section 37.31: Weak relative Noether normalization
  • Section 37.32: Bertini theorems
  • Section 37.33: Theorem of the cube
  • Section 37.34: Limit arguments
  • Section 37.35: Étale neighbourhoods
  • Section 37.36: Étale neighbourhoods and branches
  • Section 37.37: Unramified and étale morphisms
  • Section 37.38: Slicing smooth morphisms
  • Section 37.39: Étale neighbourhoods and Artin approximation
  • Section 37.40: Finite free locally dominates étale
  • Section 37.41: Étale localization of quasi-finite morphisms
  • Section 37.42: Étale localization of integral morphisms
  • Section 37.43: Zariski's Main Theorem
  • Section 37.44: Applications of Zariski's Main Theorem, I
  • Section 37.45: Applications of Zariski's Main Theorem, II
  • Section 37.46: Application to morphisms with connected fibres
  • Section 37.47: Application to the structure of finite type morphisms
  • Section 37.48: Application to the fppf topology
  • Section 37.49: Quasi-projective schemes
  • Section 37.50: Projective schemes
  • Section 37.51: Proj and Spec
  • Section 37.52: Closed points in fibres
  • Section 37.53: Stein factorization
  • Section 37.54: Generic flatness stratification
  • Section 37.55: Stratifying a morphism
  • Section 37.56: Improving morphisms of relative dimension one
  • Section 37.57: Descending separated locally quasi-finite morphisms
  • Section 37.58: Relative finite presentation
  • Section 37.59: Relative pseudo-coherence
  • Section 37.60: Pseudo-coherent morphisms
  • Section 37.61: Perfect morphisms
  • Section 37.62: Local complete intersection morphisms
  • Section 37.63: Exact sequences of differentials and conormal sheaves
  • Section 37.64: Weakly étale morphisms
  • Section 37.65: Reduced fibre theorem
  • Section 37.66: Ind-quasi-affine morphisms
  • Section 37.67: Pushouts in the category of schemes, II
  • Section 37.68: Relative morphisms
  • Section 37.69: Characterizing pseudo-coherent complexes, III
  • Section 37.70: Descent finiteness properties of complexes
  • Section 37.71: Relatively perfect objects
  • Section 37.72: Contracting rational curves
  • Section 37.73: Affine stratifications
  • Section 37.74: Universally open morphisms
  • Section 37.75: Weightings
  • Section 37.76: More on weightings
  • Section 37.77: Weightings and affine stratification numbers
  • Section 37.78: Completely decomposed morphisms
  • Section 37.79: Families of ample invertible modules
  • Section 37.80: Blowing up and ample families of invertible modules
  • Section 37.81: The extensive criterion for closed immersions
• Chapter 38: More on Flatness
  • Section 38.1: Introduction
  • Section 38.2: Lemmas on étale localization
  • Section 38.3: The local structure of a finite type module
  • Section 38.4: One step dévissage
  • Section 38.5: Complete dévissage
  • Section 38.6: Translation into algebra
  • Section 38.7: Localization and universally injective maps
  • Section 38.8: Completion and Mittag-Leffler modules
  • Section 38.9: Projective modules
  • Section 38.10: Flat finite type modules, Part I
  • Section 38.11: Extending properties from an open
  • Section 38.12: Flat finitely presented modules
  • Section 38.13: Flat finite type modules, Part II
  • Section 38.14: Examples of relatively pure modules
  • Section 38.15: Impurities
  • Section 38.16: Relatively pure modules
  • Section 38.17: Examples of relatively pure sheaves
  • Section 38.18: A criterion for purity
  • Section 38.19: How purity is used
  • Section 38.20: Flattening functors
  • Section 38.21: Flattening stratifications
  • Section 38.22: Flattening stratification over an Artinian ring
  • Section 38.23: Flattening a map
  • Section 38.24: Flattening in the local case
  • Section 38.25: Variants of a lemma
  • Section 38.26: Flat finite type modules, Part III
  • Section 38.27: Universal flattening
  • Section 38.28: Grothendieck's Existence Theorem, IV
  • Section 38.29: Grothendieck's Existence Theorem, V
  • Section 38.30: Blowing up and flatness
  • Section 38.31: Applications
  • Section 38.32: Compactifications
  • Section 38.33: Nagata compactification
  • Section 38.34: The h topology
  • Section 38.35: More on the h topology
  • Section 38.36: Blow up squares and the ph topology
  • Section 38.37: Almost blow up squares and the h topology
  • Section 38.38: Absolute weak normalization and h coverings
  • Section 38.39: Descent vector bundles in positive characteristic
  • Section 38.40: Blowing up complexes
  • Section 38.41: Blowing up perfect modules
  • Section 38.42: An operator introduced by Berthelot and Ogus
  • Section 38.43: Blowing up complexes, II
  • Section 38.44: Blowing up complexes, III
• Chapter 39: Groupoid Schemes
  • Section 39.1: Introduction
  • Section 39.2: Notation
  • Section 39.3: Equivalence relations
  • Section 39.4: Group schemes
  • Section 39.5: Examples of group schemes
  • Section 39.6: Properties of group schemes
  • Section 39.7: Properties of group schemes over a field
  • Section 39.8: Properties of algebraic group schemes
  • Section 39.9: Abelian varieties
  • Section 39.10: Actions of group schemes
  • Section 39.11: Principal homogeneous spaces
  • Section 39.12: Equivariant quasi-coherent sheaves
  • Section 39.13: Groupoids
  • Section 39.14: Quasi-coherent sheaves on groupoids
  • Section 39.15: Colimits of quasi-coherent modules
  • Section 39.16: Groupoids and group schemes
  • Section 39.17: The stabilizer group scheme
  • Section 39.18: Restricting groupoids
  • Section 39.19: Invariant subschemes
  • Section 39.20: Quotient sheaves
  • Section 39.21: Descent in terms of groupoids
  • Section 39.22: Separation conditions
  • Section 39.23: Finite flat groupoids, affine case
  • Section 39.24: Finite flat groupoids
  • Section 39.25: Descending quasi-projective schemes
• Chapter 40: More on Groupoid Schemes
  • Section 40.1: Introduction
  • Section 40.2: Notation
  • Section 40.3: Useful diagrams
  • Section 40.4: Sheaf of differentials
  • Section 40.5: Local structure
  • Section 40.6: Properties of groupoids
  • Section 40.7: Comparing fibres
  • Section 40.8: Cohen-Macaulay presentations
  • Section 40.9: Restricting groupoids
  • Section 40.10: Properties of groupoids on fields
  • Section 40.11: Morphisms of groupoids on fields
  • Section 40.12: Slicing groupoids
  • Section 40.13: Étale localization of groupoids
  • Section 40.14: Finite groupoids
  • Section 40.15: Descending ind-quasi-affine morphisms
• Chapter 41: Étale Morphisms of Schemes
  • Section 41.1: Introduction
  • Section 41.2: Conventions
  • Section 41.3: Unramified morphisms
  • Section 41.4: Three other characterizations of unramified morphisms
  • Section 41.5: The functorial characterization of unramified morphisms
  • Section 41.6: Topological properties of unramified morphisms
  • Section 41.7: Universally injective, unramified morphisms
  • Section 41.8: Examples of unramified morphisms
  • Section 41.9: Flat morphisms
  • Section 41.10: Topological properties of flat morphisms
  • Section 41.11: Étale morphisms
  • Section 41.12: The structure theorem
  • Section 41.13: Étale and smooth morphisms
  • Section 41.14: Topological properties of étale morphisms
  • Section 41.15: Topological invariance of the étale topology
  • Section 41.16: The functorial characterization
  • Section 41.17: Étale local structure of unramified morphisms
  • Section 41.18: Étale local structure of étale morphisms
  • Section 41.19: Permanence properties
  • Section 41.20: Descending étale morphisms
  • Section 41.21: Normal crossings divisors