The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

2 Schemes

  • Chapter 25: Schemes
    • Section 25.1: Introduction
    • Section 25.2: Locally ringed spaces
    • Section 25.3: Open immersions of locally ringed spaces
    • Section 25.4: Closed immersions of locally ringed spaces
    • Section 25.5: Affine schemes
    • Section 25.6: The category of affine schemes
    • Section 25.7: Quasi-coherent sheaves on affines
    • Section 25.8: Closed subspaces of affine schemes
    • Section 25.9: Schemes
    • Section 25.10: Immersions of schemes
    • Section 25.11: Zariski topology of schemes
    • Section 25.12: Reduced schemes
    • Section 25.13: Points of schemes
    • Section 25.14: Glueing schemes
    • Section 25.15: A representability criterion
    • Section 25.16: Existence of fibre products of schemes
    • Section 25.17: Fibre products of schemes
    • Section 25.18: Base change in algebraic geometry
    • Section 25.19: Quasi-compact morphisms
    • Section 25.20: Valuative criterion for universal closedness
    • Section 25.21: Separation axioms
    • Section 25.22: Valuative criterion of separatedness
    • Section 25.23: Monomorphisms
    • Section 25.24: Functoriality for quasi-coherent modules
  • Chapter 26: Constructions of Schemes
    • Section 26.1: Introduction
    • Section 26.2: Relative glueing
    • Section 26.3: Relative spectrum via glueing
    • Section 26.4: Relative spectrum as a functor
    • Section 26.5: Affine n-space
    • Section 26.6: Vector bundles
    • Section 26.7: Cones
    • Section 26.8: Proj of a graded ring
    • Section 26.9: Quasi-coherent sheaves on Proj
    • Section 26.10: Invertible sheaves on Proj
    • Section 26.11: Functoriality of Proj
    • Section 26.12: Morphisms into Proj
    • Section 26.13: Projective space
    • Section 26.14: Invertible sheaves and morphisms into Proj
    • Section 26.15: Relative Proj via glueing
    • Section 26.16: Relative Proj as a functor
    • Section 26.17: Quasi-coherent sheaves on relative Proj
    • Section 26.18: Functoriality of relative Proj
    • Section 26.19: Invertible sheaves and morphisms into relative Proj
    • Section 26.20: Twisting by invertible sheaves and relative Proj
    • Section 26.21: Projective bundles
    • Section 26.22: Grassmannians
  • Chapter 27: Properties of Schemes
    • Section 27.1: Introduction
    • Section 27.2: Constructible sets
    • Section 27.3: Integral, irreducible, and reduced schemes
    • Section 27.4: Types of schemes defined by properties of rings
    • Section 27.5: Noetherian schemes
    • Section 27.6: Jacobson schemes
    • Section 27.7: Normal schemes
    • Section 27.8: Cohen-Macaulay schemes
    • Section 27.9: Regular schemes
    • Section 27.10: Dimension
    • Section 27.11: Catenary schemes
    • Section 27.12: Serre's conditions
    • Section 27.13: Japanese and Nagata schemes
    • Section 27.14: The singular locus
    • Section 27.15: Local irreducibility
    • Section 27.16: Characterizing modules of finite type and finite presentation
    • Section 27.17: Sections over principal opens
    • Section 27.18: Quasi-affine schemes
    • Section 27.19: Flat modules
    • Section 27.20: Locally free modules
    • Section 27.21: Locally projective modules
    • Section 27.22: Extending quasi-coherent sheaves
    • Section 27.23: Gabber's result
    • Section 27.24: Sections with support in a closed subset
    • Section 27.25: Sections of quasi-coherent sheaves
    • Section 27.26: Ample invertible sheaves
    • Section 27.27: Affine and quasi-affine schemes
    • Section 27.28: Quasi-coherent sheaves and ample invertible sheaves
    • Section 27.29: Finding suitable affine opens
  • Chapter 28: Morphisms of Schemes
    • Section 28.1: Introduction
    • Section 28.2: Closed immersions
    • Section 28.3: Immersions
    • Section 28.4: Closed immersions and quasi-coherent sheaves
    • Section 28.5: Supports of modules
    • Section 28.6: Scheme theoretic image
    • Section 28.7: Scheme theoretic closure and density
    • Section 28.8: Dominant morphisms
    • Section 28.9: Surjective morphisms
    • Section 28.10: Radicial and universally injective morphisms
    • Section 28.11: Affine morphisms
    • Section 28.12: Quasi-affine morphisms
    • Section 28.13: Types of morphisms defined by properties of ring maps
    • Section 28.14: Morphisms of finite type
    • Section 28.15: Points of finite type and Jacobson schemes
    • Section 28.16: Universally catenary schemes
    • Section 28.17: Nagata schemes, reprise
    • Section 28.18: The singular locus, reprise
    • Section 28.19: Quasi-finite morphisms
    • Section 28.20: Morphisms of finite presentation
    • Section 28.21: Constructible sets
    • Section 28.22: Open morphisms
    • Section 28.23: Submersive morphisms
    • Section 28.24: Flat morphisms
    • Section 28.25: Flat closed immersions
    • Section 28.26: Generic flatness
    • Section 28.27: Morphisms and dimensions of fibres
    • Section 28.28: Morphisms of given relative dimension
    • Section 28.29: Syntomic morphisms
    • Section 28.30: Conormal sheaf of an immersion
    • Section 28.31: Sheaf of differentials of a morphism
    • Section 28.32: Smooth morphisms
    • Section 28.33: Unramified morphisms
    • Section 28.34: Étale morphisms
    • Section 28.35: Relatively ample sheaves
    • Section 28.36: Very ample sheaves
    • Section 28.37: Ample and very ample sheaves relative to finite type morphisms
    • Section 28.38: Quasi-projective morphisms
    • Section 28.39: Proper morphisms
    • Section 28.40: Valuative criteria
    • Section 28.41: Projective morphisms
    • Section 28.42: Integral and finite morphisms
    • Section 28.43: Universal homeomorphisms
    • Section 28.44: Universal homeomorphisms of affine schemes
    • Section 28.45: Absolute weak normalization and seminormalization
    • Section 28.46: Finite locally free morphisms
    • Section 28.47: Rational maps
    • Section 28.48: Birational morphisms
    • Section 28.49: Generically finite morphisms
    • Section 28.50: The dimension formula
    • Section 28.51: Relative normalization
    • Section 28.52: Normalization
    • Section 28.53: Zariski's Main Theorem (algebraic version)
    • Section 28.54: Universally bounded fibres
  • Chapter 29: Cohomology of Schemes
    • Section 29.1: Introduction
    • Section 29.2: Čech cohomology of quasi-coherent sheaves
    • Section 29.3: Vanishing of cohomology
    • Section 29.4: Quasi-coherence of higher direct images
    • Section 29.5: Cohomology and base change, I
    • Section 29.6: Colimits and higher direct images
    • Section 29.7: Cohomology and base change, II
    • Section 29.8: Cohomology of projective space
    • Section 29.9: Coherent sheaves on locally Noetherian schemes
    • Section 29.10: Coherent sheaves on Noetherian schemes
    • Section 29.11: Depth
    • Section 29.12: Devissage of coherent sheaves
    • Section 29.13: Finite morphisms and affines
    • Section 29.14: Coherent sheaves on Proj, I
    • Section 29.15: Coherent sheaves on Proj, II
    • Section 29.16: Higher direct images along projective morphisms
    • Section 29.17: Ample invertible sheaves and cohomology
    • Section 29.18: Chow's Lemma
    • Section 29.19: Higher direct images of coherent sheaves
    • Section 29.20: The theorem on formal functions
    • Section 29.21: Applications of the theorem on formal functions
    • Section 29.22: Cohomology and base change, III
    • Section 29.23: Coherent formal modules
    • Section 29.24: Grothendieck's existence theorem, I
    • Section 29.25: Grothendieck's existence theorem, II
    • Section 29.26: Being proper over a base
    • Section 29.27: Grothendieck's existence theorem, III
    • Section 29.28: Grothendieck's algebraization theorem
  • Chapter 30: Divisors
    • Section 30.1: Introduction
    • Section 30.2: Associated points
    • Section 30.3: Morphisms and associated points
    • Section 30.4: Embedded points
    • Section 30.5: Weakly associated points
    • Section 30.6: Morphisms and weakly associated points
    • Section 30.7: Relative assassin
    • Section 30.8: Relative weak assassin
    • Section 30.9: Fitting ideals
    • Section 30.10: The singular locus of a morphism
    • Section 30.11: Torsion free modules
    • Section 30.12: Reflexive modules
    • Section 30.13: Effective Cartier divisors
    • Section 30.14: Effective Cartier divisors and invertible sheaves
    • Section 30.15: Effective Cartier divisors on Noetherian schemes
    • Section 30.16: Complements of affine opens
    • Section 30.17: Norms
    • Section 30.18: Relative effective Cartier divisors
    • Section 30.19: The normal cone of an immersion
    • Section 30.20: Regular ideal sheaves
    • Section 30.21: Regular immersions
    • Section 30.22: Relative regular immersions
    • Section 30.23: Meromorphic functions and sections
    • Section 30.24: Meromorphic functions and sections; Noetherian case
    • Section 30.25: Meromorphic functions and sections; reduced case
    • Section 30.26: Weil divisors
    • Section 30.27: The Weil divisor class associated to an invertible module
    • Section 30.28: More on invertible modules
    • Section 30.29: Weil divisors on normal schemes
    • Section 30.30: Relative Proj
    • Section 30.31: Closed subschemes of relative proj
    • Section 30.32: Blowing up
    • Section 30.33: Strict transform
    • Section 30.34: Admissible blowups
    • Section 30.35: Modifications
  • Chapter 31: Limits of Schemes
    • Section 31.1: Introduction
    • Section 31.2: Directed limits of schemes with affine transition maps
    • Section 31.3: Infinite products
    • Section 31.4: Descending properties
    • Section 31.5: Absolute Noetherian Approximation
    • Section 31.6: Limits and morphisms of finite presentation
    • Section 31.7: Relative approximation
    • Section 31.8: Descending properties of morphisms
    • Section 31.9: Finite type closed in finite presentation
    • Section 31.10: Descending relative objects
    • Section 31.11: Characterizing affine schemes
    • Section 31.12: Variants of Chow's Lemma
    • Section 31.13: Applications of Chow's lemma
    • Section 31.14: Universally closed morphisms
    • Section 31.15: Noetherian valuative criterion
    • Section 31.16: Limits and dimensions of fibres
    • Section 31.17: Base change in top degree
    • Section 31.18: Glueing in closed fibres
    • Section 31.19: Application to modifications
    • Section 31.20: Descending finite type schemes
  • Chapter 32: Varieties
    • Section 32.1: Introduction
    • Section 32.2: Notation
    • Section 32.3: Varieties
    • Section 32.4: Varieties and rational maps
    • Section 32.5: Change of fields and local rings
    • Section 32.6: Geometrically reduced schemes
    • Section 32.7: Geometrically connected schemes
    • Section 32.8: Geometrically irreducible schemes
    • Section 32.9: Geometrically integral schemes
    • Section 32.10: Geometrically normal schemes
    • Section 32.11: Change of fields and locally Noetherian schemes
    • Section 32.12: Geometrically regular schemes
    • Section 32.13: Change of fields and the Cohen-Macaulay property
    • Section 32.14: Change of fields and the Jacobson property
    • Section 32.15: Change of fields and ample invertible sheaves
    • Section 32.16: Tangent spaces
    • Section 32.17: Generically finite morphisms
    • Section 32.18: Variants of Noether normalization
    • Section 32.19: Dimension of fibres
    • Section 32.20: Algebraic schemes
    • Section 32.21: Complete local rings
    • Section 32.22: Global generation
    • Section 32.23: Separating points and tangent vectors
    • Section 32.24: Closures of products
    • Section 32.25: Schemes smooth over fields
    • Section 32.26: Types of varieties
    • Section 32.27: Normalization
    • Section 32.28: Groups of invertible functions
    • Section 32.29: Künneth formula
    • Section 32.30: Picard groups of varieties
    • Section 32.31: Uniqueness of base field
    • Section 32.32: Euler characteristics
    • Section 32.33: Projective space
    • Section 32.34: Coherent sheaves on projective space
    • Section 32.35: Frobenii
    • Section 32.36: Glueing dimension one rings
    • Section 32.37: One dimensional Noetherian schemes
    • Section 32.38: The delta invariant
    • Section 32.39: The number of branches
    • Section 32.40: Normalization of one dimensional schemes
    • Section 32.41: Finding affine opens
    • Section 32.42: Curves
    • Section 32.43: Degrees on curves
    • Section 32.44: Numerical intersections
    • Section 32.45: Embedding dimension
  • Chapter 33: Topologies on Schemes
    • Section 33.1: Introduction
    • Section 33.2: The general procedure
    • Section 33.3: The Zariski topology
    • Section 33.4: The étale topology
    • Section 33.5: The smooth topology
    • Section 33.6: The syntomic topology
    • Section 33.7: The fppf topology
    • Section 33.8: The ph topology
    • Section 33.9: The fpqc topology
    • Section 33.10: The V topology
    • Section 33.11: Change of topologies
    • Section 33.12: Change of big sites
    • Section 33.13: Extending functors
  • Chapter 34: Descent
    • Section 34.1: Introduction
    • Section 34.2: Descent data for quasi-coherent sheaves
    • Section 34.3: Descent for modules
    • Section 34.4: Descent for universally injective morphisms
    • Section 34.5: Fpqc descent of quasi-coherent sheaves
    • Section 34.6: Galois descent for quasi-coherent sheaves
    • Section 34.7: Descent of finiteness properties of modules
    • Section 34.8: Quasi-coherent sheaves and topologies
    • Section 34.9: Parasitic modules
    • Section 34.10: Fpqc coverings are universal effective epimorphisms
    • Section 34.11: Descent of finiteness and smoothness properties of morphisms
    • Section 34.12: Local properties of schemes
    • Section 34.13: Properties of schemes local in the fppf topology
    • Section 34.14: Properties of schemes local in the syntomic topology
    • Section 34.15: Properties of schemes local in the smooth topology
    • Section 34.16: Variants on descending properties
    • Section 34.17: Germs of schemes
    • Section 34.18: Local properties of germs
    • Section 34.19: Properties of morphisms local on the target
    • Section 34.20: Properties of morphisms local in the fpqc topology on the target
    • Section 34.21: Properties of morphisms local in the fppf topology on the target
    • Section 34.22: Application of fpqc descent of properties of morphisms
    • Section 34.23: Properties of morphisms local on the source
    • Section 34.24: Properties of morphisms local in the fpqc topology on the source
    • Section 34.25: Properties of morphisms local in the fppf topology on the source
    • Section 34.26: Properties of morphisms local in the syntomic topology on the source
    • Section 34.27: Properties of morphisms local in the smooth topology on the source
    • Section 34.28: Properties of morphisms local in the étale topology on the source
    • Section 34.29: Properties of morphisms étale local on source-and-target
    • Section 34.30: Properties of morphisms of germs local on source-and-target
    • Section 34.31: Descent data for schemes over schemes
    • Section 34.32: Fully faithfulness of the pullback functors
    • Section 34.33: Descending types of morphisms
    • Section 34.34: Descending affine morphisms
    • Section 34.35: Descending quasi-affine morphisms
    • Section 34.36: Descent data in terms of sheaves
  • Chapter 35: Derived Categories of Schemes
    • Section 35.1: Introduction
    • Section 35.2: Conventions
    • Section 35.3: Derived category of quasi-coherent modules
    • Section 35.4: Total direct image
    • Section 35.5: Affine morphisms
    • Section 35.6: The coherator
    • Section 35.7: The coherator for Noetherian schemes
    • Section 35.8: Koszul complexes
    • Section 35.9: Pseudo-coherent and perfect complexes
    • Section 35.10: Derived category of coherent modules
    • Section 35.11: Descent finiteness properties of complexes
    • Section 35.12: Lifting complexes
    • Section 35.13: Approximation by perfect complexes
    • Section 35.14: Generating derived categories
    • Section 35.15: An example generator
    • Section 35.16: Compact and perfect objects
    • Section 35.17: Derived categories as module categories
    • Section 35.18: Characterizing pseudo-coherent complexes, I
    • Section 35.19: An example equivalence
    • Section 35.20: The coherator revisited
    • Section 35.21: Cohomology and base change, IV
    • Section 35.22: Cohomology and base change, V
    • Section 35.23: Producing perfect complexes
    • Section 35.24: A projection formula for Ext
    • Section 35.25: Limits and derived categories
    • Section 35.26: Cohomology and base change, VI
    • Section 35.27: Perfect complexes
    • Section 35.28: Applications
    • Section 35.29: Other applications
    • Section 35.30: Characterizing pseudo-coherent complexes, II
    • Section 35.31: Relatively perfect objects
  • Chapter 36: More on Morphisms
    • Section 36.1: Introduction
    • Section 36.2: Thickenings
    • Section 36.3: Morphisms of thickenings
    • Section 36.4: Picard groups of thickenings
    • Section 36.5: First order infinitesimal neighbourhood
    • Section 36.6: Formally unramified morphisms
    • Section 36.7: Universal first order thickenings
    • Section 36.8: Formally étale morphisms
    • Section 36.9: Infinitesimal deformations of maps
    • Section 36.10: Infinitesimal deformations of schemes
    • Section 36.11: Formally smooth morphisms
    • Section 36.12: Smoothness over a Noetherian base
    • Section 36.13: The naive cotangent complex
    • Section 36.14: Pushouts in the category of schemes, I
    • Section 36.15: Openness of the flat locus
    • Section 36.16: Critère de platitude par fibres
    • Section 36.17: Normalization revisited
    • Section 36.18: Normal morphisms
    • Section 36.19: Regular morphisms
    • Section 36.20: Cohen-Macaulay morphisms
    • Section 36.21: Slicing Cohen-Macaulay morphisms
    • Section 36.22: Generic fibres
    • Section 36.23: Relative assassins
    • Section 36.24: Reduced fibres
    • Section 36.25: Irreducible components of fibres
    • Section 36.26: Connected components of fibres
    • Section 36.27: Connected components meeting a section
    • Section 36.28: Dimension of fibres
    • Section 36.29: Theorem of the cube
    • Section 36.30: Limit arguments
    • Section 36.31: Étale neighbourhoods
    • Section 36.32: Étale neighbourhoods and Artin approximation
    • Section 36.33: Étale neighbourhoods and branches
    • Section 36.34: Slicing smooth morphisms
    • Section 36.35: Finite free locally dominates étale
    • Section 36.36: Étale localization of quasi-finite morphisms
    • Section 36.37: Étale localization of integral morphisms
    • Section 36.38: Zariski's Main Theorem
    • Section 36.39: Application to morphisms with connected fibres
    • Section 36.40: Application to the structure of finite type morphisms
    • Section 36.41: Application to the fppf topology
    • Section 36.42: Quasi-projective schemes
    • Section 36.43: Projective schemes
    • Section 36.44: Proj and Spec
    • Section 36.45: Closed points in fibres
    • Section 36.46: Stein factorization
    • Section 36.47: Descending separated locally quasi-finite morphisms
    • Section 36.48: Relative finite presentation
    • Section 36.49: Relative pseudo-coherence
    • Section 36.50: Pseudo-coherent morphisms
    • Section 36.51: Perfect morphisms
    • Section 36.52: Local complete intersection morphisms
    • Section 36.53: Exact sequences of differentials and conormal sheaves
    • Section 36.54: Weakly étale morphisms
    • Section 36.55: Reduced fibre theorem
    • Section 36.56: Ind-quasi-affine morphisms
    • Section 36.57: Pushouts in the category of schemes, II
    • Section 36.58: Relative morphisms
    • Section 36.59: Characterizing pseudo-coherent complexes, III
    • Section 36.60: Descent finiteness properties of complexes
    • Section 36.61: Relatively perfect objects
    • Section 36.62: Contracting rational curves
  • Chapter 37: More on Flatness
    • Section 37.1: Introduction
    • Section 37.2: Lemmas on étale localization
    • Section 37.3: The local structure of a finite type module
    • Section 37.4: One step dévissage
    • Section 37.5: Complete dévissage
    • Section 37.6: Translation into algebra
    • Section 37.7: Localization and universally injective maps
    • Section 37.8: Completion and Mittag-Leffler modules
    • Section 37.9: Projective modules
    • Section 37.10: Flat finite type modules, Part I
    • Section 37.11: Extending properties from an open
    • Section 37.12: Flat finitely presented modules
    • Section 37.13: Flat finite type modules, Part II
    • Section 37.14: Examples of relatively pure modules
    • Section 37.15: Impurities
    • Section 37.16: Relatively pure modules
    • Section 37.17: Examples of relatively pure sheaves
    • Section 37.18: A criterion for purity
    • Section 37.19: How purity is used
    • Section 37.20: Flattening functors
    • Section 37.21: Flattening stratifications
    • Section 37.22: Flattening stratification over an Artinian ring
    • Section 37.23: Flattening a map
    • Section 37.24: Flattening in the local case
    • Section 37.25: Variants of a lemma
    • Section 37.26: Flat finite type modules, Part III
    • Section 37.27: Universal flattening
    • Section 37.28: Grothendieck's Existence Theorem, IV
    • Section 37.29: Grothendieck's Existence Theorem, V
    • Section 37.30: Blowing up and flatness
    • Section 37.31: Applications
    • Section 37.32: The h topology
    • Section 37.33: More on the h topology
    • Section 37.34: Blow up squares and the ph topology
    • Section 37.35: Almost blow up squares and the h topology
    • Section 37.36: Absolute weak normalization and h coverings
    • Section 37.37: Descent vector bundles in positive characteristic
    • Section 37.38: Blowing up complexes
  • Chapter 38: Groupoid Schemes
    • Section 38.1: Introduction
    • Section 38.2: Notation
    • Section 38.3: Equivalence relations
    • Section 38.4: Group schemes
    • Section 38.5: Examples of group schemes
    • Section 38.6: Properties of group schemes
    • Section 38.7: Properties of group schemes over a field
    • Section 38.8: Properties of algebraic group schemes
    • Section 38.9: Abelian varieties
    • Section 38.10: Actions of group schemes
    • Section 38.11: Principal homogeneous spaces
    • Section 38.12: Equivariant quasi-coherent sheaves
    • Section 38.13: Groupoids
    • Section 38.14: Quasi-coherent sheaves on groupoids
    • Section 38.15: Colimits of quasi-coherent modules
    • Section 38.16: Groupoids and group schemes
    • Section 38.17: The stabilizer group scheme
    • Section 38.18: Restricting groupoids
    • Section 38.19: Invariant subschemes
    • Section 38.20: Quotient sheaves
    • Section 38.21: Descent in terms of groupoids
    • Section 38.22: Separation conditions
    • Section 38.23: Finite flat groupoids, affine case
    • Section 38.24: Finite flat groupoids
    • Section 38.25: Descending quasi-projective schemes
  • Chapter 39: More on Groupoid Schemes
    • Section 39.1: Introduction
    • Section 39.2: Notation
    • Section 39.3: Useful diagrams
    • Section 39.4: Sheaf of differentials
    • Section 39.5: Local structure
    • Section 39.6: Properties of groupoids
    • Section 39.7: Comparing fibres
    • Section 39.8: Cohen-Macaulay presentations
    • Section 39.9: Restricting groupoids
    • Section 39.10: Properties of groupoids on fields
    • Section 39.11: Morphisms of groupoids on fields
    • Section 39.12: Slicing groupoids
    • Section 39.13: Étale localization of groupoids
    • Section 39.14: Finite groupoids
    • Section 39.15: Descending ind-quasi-affine morphisms
  • Chapter 40: Étale Morphisms of Schemes
    • Section 40.1: Introduction
    • Section 40.2: Conventions
    • Section 40.3: Unramified morphisms
    • Section 40.4: Three other characterizations of unramified morphisms
    • Section 40.5: The functorial characterization of unramified morphisms
    • Section 40.6: Topological properties of unramified morphisms
    • Section 40.7: Universally injective, unramified morphisms
    • Section 40.8: Examples of unramified morphisms
    • Section 40.9: Flat morphisms
    • Section 40.10: Topological properties of flat morphisms
    • Section 40.11: Étale morphisms
    • Section 40.12: The structure theorem
    • Section 40.13: Étale and smooth morphisms
    • Section 40.14: Topological properties of étale morphisms
    • Section 40.15: Topological invariance of the étale topology
    • Section 40.16: The functorial characterization
    • Section 40.17: Étale local structure of unramified morphisms
    • Section 40.18: Étale local structure of étale morphisms
    • Section 40.19: Permanence properties
    • Section 40.20: Descending étale morphisms
    • Section 40.21: Normal crossings divisors