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*}

4 Algebraic Spaces

  • Chapter 57: Algebraic Spaces
    • Section 57.1: Introduction
    • Section 57.2: General remarks
    • Section 57.3: Representable morphisms of presheaves
    • Section 57.4: Lists of useful properties of morphisms of schemes
    • Section 57.5: Properties of representable morphisms of presheaves
    • Section 57.6: Algebraic spaces
    • Section 57.7: Fibre products of algebraic spaces
    • Section 57.8: Glueing algebraic spaces
    • Section 57.9: Presentations of algebraic spaces
    • Section 57.10: Algebraic spaces and equivalence relations
    • Section 57.11: Algebraic spaces, retrofitted
    • Section 57.12: Immersions and Zariski coverings of algebraic spaces
    • Section 57.13: Separation conditions on algebraic spaces
    • Section 57.14: Examples of algebraic spaces
    • Section 57.15: Change of big site
    • Section 57.16: Change of base scheme
  • Chapter 58: Properties of Algebraic Spaces
    • Section 58.1: Introduction
    • Section 58.2: Conventions
    • Section 58.3: Separation axioms
    • Section 58.4: Points of algebraic spaces
    • Section 58.5: Quasi-compact spaces
    • Section 58.6: Special coverings
    • Section 58.7: Properties of Spaces defined by properties of schemes
    • Section 58.8: Constructible sets
    • Section 58.9: Dimension at a point
    • Section 58.10: Dimension of local rings
    • Section 58.11: Generic points
    • Section 58.12: Reduced spaces
    • Section 58.13: The schematic locus
    • Section 58.14: Obtaining a scheme
    • Section 58.15: Points on quasi-separated spaces
    • Section 58.16: Étale morphisms of algebraic spaces
    • Section 58.17: Spaces and fpqc coverings
    • Section 58.18: The étale site of an algebraic space
    • Section 58.19: Points of the small étale site
    • Section 58.20: Supports of abelian sheaves
    • Section 58.21: The structure sheaf of an algebraic space
    • Section 58.22: Stalks of the structure sheaf
    • Section 58.23: Local irreducibility
    • Section 58.24: Noetherian spaces
    • Section 58.25: Regular algebraic spaces
    • Section 58.26: Sheaves of modules on algebraic spaces
    • Section 58.27: Étale localization
    • Section 58.28: Recovering morphisms
    • Section 58.29: Quasi-coherent sheaves on algebraic spaces
    • Section 58.30: Properties of modules
    • Section 58.31: Locally projective modules
    • Section 58.32: Quasi-coherent sheaves and presentations
    • Section 58.33: Morphisms towards schemes
    • Section 58.34: Quotients by free actions
  • Chapter 59: Morphisms of Algebraic Spaces
    • Section 59.1: Introduction
    • Section 59.2: Conventions
    • Section 59.3: Properties of representable morphisms
    • Section 59.4: Separation axioms
    • Section 59.5: Surjective morphisms
    • Section 59.6: Open morphisms
    • Section 59.7: Submersive morphisms
    • Section 59.8: Quasi-compact morphisms
    • Section 59.9: Universally closed morphisms
    • Section 59.10: Monomorphisms
    • Section 59.11: Pushforward of quasi-coherent sheaves
    • Section 59.12: Immersions
    • Section 59.13: Closed immersions
    • Section 59.14: Closed immersions and quasi-coherent sheaves
    • Section 59.15: Supports of modules
    • Section 59.16: Scheme theoretic image
    • Section 59.17: Scheme theoretic closure and density
    • Section 59.18: Dominant morphisms
    • Section 59.19: Universally injective morphisms
    • Section 59.20: Affine morphisms
    • Section 59.21: Quasi-affine morphisms
    • Section 59.22: Types of morphisms étale local on source-and-target
    • Section 59.23: Morphisms of finite type
    • Section 59.24: Points and geometric points
    • Section 59.25: Points of finite type
    • Section 59.26: Nagata spaces
    • Section 59.27: Quasi-finite morphisms
    • Section 59.28: Morphisms of finite presentation
    • Section 59.29: Constructible sets
    • Section 59.30: Flat morphisms
    • Section 59.31: Flat modules
    • Section 59.32: Generic flatness
    • Section 59.33: Relative dimension
    • Section 59.34: Morphisms and dimensions of fibres
    • Section 59.35: The dimension formula
    • Section 59.36: Syntomic morphisms
    • Section 59.37: Smooth morphisms
    • Section 59.38: Unramified morphisms
    • Section 59.39: Étale morphisms
    • Section 59.40: Proper morphisms
    • Section 59.41: Valuative criteria
    • Section 59.42: Valuative criterion for universal closedness
    • Section 59.43: Valuative criterion of separatedness
    • Section 59.44: Valuative criterion of properness
    • Section 59.45: Integral and finite morphisms
    • Section 59.46: Finite locally free morphisms
    • Section 59.47: Rational maps
    • Section 59.48: Relative normalization of algebraic spaces
    • Section 59.49: Normalization
    • Section 59.50: Separated, locally quasi-finite morphisms
    • Section 59.51: Applications
    • Section 59.52: Zariski's Main Theorem (representable case)
    • Section 59.53: Universal homeomorphisms
  • Chapter 60: Decent Algebraic Spaces
    • Section 60.1: Introduction
    • Section 60.2: Conventions
    • Section 60.3: Universally bounded fibres
    • Section 60.4: Finiteness conditions and points
    • Section 60.5: Conditions on algebraic spaces
    • Section 60.6: Reasonable and decent algebraic spaces
    • Section 60.7: Points and specializations
    • Section 60.8: Stratifying algebraic spaces by schemes
    • Section 60.9: Integral cover by a scheme
    • Section 60.10: Schematic locus
    • Section 60.11: Residue fields and henselian local rings
    • Section 60.12: Points on decent spaces
    • Section 60.13: Reduced singleton spaces
    • Section 60.14: Decent spaces
    • Section 60.15: Locally separated spaces
    • Section 60.16: Valuative criterion
    • Section 60.17: Relative conditions
    • Section 60.18: Points of fibres
    • Section 60.19: Monomorphisms
    • Section 60.20: Generic points
    • Section 60.21: Generically finite morphisms
    • Section 60.22: Birational morphisms
    • Section 60.23: Jacobson spaces
    • Section 60.24: Local irreducibility
    • Section 60.25: Catenary algebraic spaces
  • Chapter 61: Cohomology of Algebraic Spaces
    • Section 61.1: Introduction
    • Section 61.2: Conventions
    • Section 61.3: Higher direct images
    • Section 61.4: Finite morphisms
    • Section 61.5: Colimits and cohomology
    • Section 61.6: The alternating Čech complex
    • Section 61.7: Higher vanishing for quasi-coherent sheaves
    • Section 61.8: Vanishing for higher direct images
    • Section 61.9: Cohomology with support in a closed subspace
    • Section 61.10: Vanishing above the dimension
    • Section 61.11: Cohomology and base change, I
    • Section 61.12: Coherent modules on locally Noetherian algebraic spaces
    • Section 61.13: Coherent sheaves on Noetherian spaces
    • Section 61.14: Devissage of coherent sheaves
    • Section 61.15: Limits of coherent modules
    • Section 61.16: Vanishing of cohomology
    • Section 61.17: Finite morphisms and affines
    • Section 61.18: A weak version of Chow's lemma
    • Section 61.19: Noetherian valuative criterion
    • Section 61.20: Higher direct images of coherent sheaves
    • Section 61.21: The theorem on formal functions
    • Section 61.22: Applications of the theorem on formal functions
  • Chapter 62: Limits of Algebraic Spaces
    • Section 62.1: Introduction
    • Section 62.2: Conventions
    • Section 62.3: Morphisms of finite presentation
    • Section 62.4: Limits of algebraic spaces
    • Section 62.5: Descending properties
    • Section 62.6: Descending properties of morphisms
    • Section 62.7: Descending relative objects
    • Section 62.8: Absolute Noetherian approximation
    • Section 62.9: Applications
    • Section 62.10: Relative approximation
    • Section 62.11: Finite type closed in finite presentation
    • Section 62.12: Approximating proper morphisms
    • Section 62.13: Embedding into affine space
    • Section 62.14: Sections with support in a closed subset
    • Section 62.15: Characterizing affine spaces
    • Section 62.16: Finite cover by a scheme
    • Section 62.17: Obtaining schemes
    • Section 62.18: Glueing in closed fibres
    • Section 62.19: Application to modifications
    • Section 62.20: Universally closed morphisms
    • Section 62.21: Noetherian valuative criterion
    • Section 62.22: Descending finite type spaces
  • Chapter 63: Divisors on Algebraic Spaces
    • Section 63.1: Introduction
    • Section 63.2: Associated and weakly associated points
    • Section 63.3: Morphisms and weakly associated points
    • Section 63.4: Relative weak assassin
    • Section 63.5: Fitting ideals
    • Section 63.6: Effective Cartier divisors
    • Section 63.7: Effective Cartier divisors and invertible sheaves
    • Section 63.8: Effective Cartier divisors on Noetherian spaces
    • Section 63.9: Relative effective Cartier divisors
    • Section 63.10: Meromorphic functions and sections
    • Section 63.11: Relative Proj
    • Section 63.12: Functoriality of relative proj
    • Section 63.13: Invertible sheaves and morphisms into relative Proj
    • Section 63.14: Relatively ample sheaves
    • Section 63.15: Relative ampleness and cohomology
    • Section 63.16: Closed subspaces of relative proj
    • Section 63.17: Blowing up
    • Section 63.18: Strict transform
    • Section 63.19: Admissible blowups
  • Chapter 64: Algebraic Spaces over Fields
    • Section 64.1: Introduction
    • Section 64.2: Conventions
    • Section 64.3: Generically finite morphisms
    • Section 64.4: Integral algebraic spaces
    • Section 64.5: Morphisms between integral algebraic spaces
    • Section 64.6: Weil divisors
    • Section 64.7: The Weil divisor class associated to an invertible module
    • Section 64.8: Modifications and alterations
    • Section 64.9: Schematic locus
    • Section 64.10: Schematic locus and field extension
    • Section 64.11: Geometrically reduced algebraic spaces
    • Section 64.12: Geometrically connected algebraic spaces
    • Section 64.13: Geometrically irreducible algebraic spaces
    • Section 64.14: Geometrically integral algebraic spaces
    • Section 64.15: Dimension
    • Section 64.16: Spaces smooth over fields
    • Section 64.17: Euler characteristics
    • Section 64.18: Numerical intersections
  • Chapter 65: Topologies on Algebraic Spaces
    • Section 65.1: Introduction
    • Section 65.2: The general procedure
    • Section 65.3: Zariski topology
    • Section 65.4: Étale topology
    • Section 65.5: Smooth topology
    • Section 65.6: Syntomic topology
    • Section 65.7: Fppf topology
    • Section 65.8: The ph topology
    • Section 65.9: Fpqc topology
  • Chapter 66: Descent and Algebraic Spaces
    • Section 66.1: Introduction
    • Section 66.2: Conventions
    • Section 66.3: Descent data for quasi-coherent sheaves
    • Section 66.4: Fpqc descent of quasi-coherent sheaves
    • Section 66.5: Descent of finiteness properties of modules
    • Section 66.6: Fpqc coverings
    • Section 66.7: Descent of finiteness and smoothness properties of morphisms
    • Section 66.8: Descending properties of spaces
    • Section 66.9: Descending properties of morphisms
    • Section 66.10: Descending properties of morphisms in the fpqc topology
    • Section 66.11: Descending properties of morphisms in the fppf topology
    • Section 66.12: Application of descent of properties of morphisms
    • Section 66.13: Properties of morphisms local on the source
    • Section 66.14: Properties of morphisms local in the fpqc topology on the source
    • Section 66.15: Properties of morphisms local in the fppf topology on the source
    • Section 66.16: Properties of morphisms local in the syntomic topology on the source
    • Section 66.17: Properties of morphisms local in the smooth topology on the source
    • Section 66.18: Properties of morphisms local in the étale topology on the source
    • Section 66.19: Properties of morphisms smooth local on source-and-target
    • Section 66.20: Properties of morphisms étale-smooth local on source-and-target
    • Section 66.21: Descent data for spaces over spaces
    • Section 66.22: Descent data in terms of sheaves
  • Chapter 67: Derived Categories of Spaces
    • Section 67.1: Introduction
    • Section 67.2: Conventions
    • Section 67.3: Generalities
    • Section 67.4: Derived category of quasi-coherent modules on the small étale site
    • Section 67.5: Derived category of quasi-coherent modules
    • Section 67.6: Total direct image
    • Section 67.7: Being proper over a base
    • Section 67.8: Derived category of coherent modules
    • Section 67.9: Induction principle
    • Section 67.10: Mayer-Vietoris
    • Section 67.11: The coherator
    • Section 67.12: The coherator for Noetherian spaces
    • Section 67.13: Pseudo-coherent and perfect complexes
    • Section 67.14: Approximation by perfect complexes
    • Section 67.15: Generating derived categories
    • Section 67.16: Compact and perfect objects
    • Section 67.17: Derived categories as module categories
    • Section 67.18: Characterizing pseudo-coherent complexes, I
    • Section 67.19: The coherator revisited
    • Section 67.20: Cohomology and base change, IV
    • Section 67.21: Cohomology and base change, V
    • Section 67.22: Producing perfect complexes
    • Section 67.23: A projection formula for Ext
    • Section 67.24: Limits and derived categories
    • Section 67.25: Cohomology and base change, VI
    • Section 67.26: Perfect complexes
    • Section 67.27: Other applications
  • Chapter 68: More on Morphisms of Spaces
    • Section 68.1: Introduction
    • Section 68.2: Conventions
    • Section 68.3: Radicial morphisms
    • Section 68.4: Monomorphisms
    • Section 68.5: Conormal sheaf of an immersion
    • Section 68.6: The normal cone of an immersion
    • Section 68.7: Sheaf of differentials of a morphism
    • Section 68.8: Topological invariance of the étale site
    • Section 68.9: Thickenings
    • Section 68.10: Morphisms of thickenings
    • Section 68.11: Picard groups of thickenings
    • Section 68.12: First order infinitesimal neighbourhood
    • Section 68.13: Formally smooth, étale, unramified transformations
    • Section 68.14: Formally unramified morphisms
    • Section 68.15: Universal first order thickenings
    • Section 68.16: Formally étale morphisms
    • Section 68.17: Infinitesimal deformations of maps
    • Section 68.18: Infinitesimal deformations of algebraic spaces
    • Section 68.19: Formally smooth morphisms
    • Section 68.20: Smoothness over a Noetherian base
    • Section 68.21: The naive cotangent complex
    • Section 68.22: Openness of the flat locus
    • Section 68.23: Critère de platitude par fibres
    • Section 68.24: Flatness over a Noetherian base
    • Section 68.25: Normalization revisited
    • Section 68.26: Cohen-Macaulay morphisms
    • Section 68.27: Gorenstein morphisms
    • Section 68.28: Slicing Cohen-Macaulay morphisms
    • Section 68.29: Reduced fibres
    • Section 68.30: Connected components of fibres
    • Section 68.31: Dimension of fibres
    • Section 68.32: Catenary algebraic spaces
    • Section 68.33: Étale localization of morphisms
    • Section 68.34: Zariski's Main Theorem
    • Section 68.35: Stein factorization
    • Section 68.36: Extending properties from an open
    • Section 68.37: Blowing up and flatness
    • Section 68.38: Applications
    • Section 68.39: Chow's lemma
    • Section 68.40: Variants of Chow's Lemma
    • Section 68.41: Grothendieck's existence theorem
    • Section 68.42: Grothendieck's algebraization theorem
    • Section 68.43: Regular immersions
    • Section 68.44: Relative pseudo-coherence
    • Section 68.45: Pseudo-coherent morphisms
    • Section 68.46: Perfect morphisms
    • Section 68.47: Local complete intersection morphisms
    • Section 68.48: When is a morphism an isomorphism?
    • Section 68.49: Exact sequences of differentials and conormal sheaves
    • Section 68.50: Characterizing pseudo-coherent complexes, II
    • Section 68.51: Relatively perfect objects
    • Section 68.52: Theorem of the cube
    • Section 68.53: Descent of finiteness properties of complexes
    • Section 68.54: Families of nodal curves
  • Chapter 69: Flatness on Algebraic Spaces
    • Section 69.1: Introduction
    • Section 69.2: Impurities
    • Section 69.3: Relatively pure modules
    • Section 69.4: Flat finite type modules
    • Section 69.5: Flat finitely presented modules
    • Section 69.6: A criterion for purity
    • Section 69.7: Flattening functors
    • Section 69.8: Making a map zero
    • Section 69.9: Flattening a map
    • Section 69.10: Flattening in the local case
    • Section 69.11: Universal flattening
    • Section 69.12: Grothendieck's Existence Theorem
    • Section 69.13: Grothendieck's Existence Theorem, bis
  • Chapter 70: Groupoids in Algebraic Spaces
    • Section 70.1: Introduction
    • Section 70.2: Conventions
    • Section 70.3: Notation
    • Section 70.4: Equivalence relations
    • Section 70.5: Group algebraic spaces
    • Section 70.6: Properties of group algebraic spaces
    • Section 70.7: Examples of group algebraic spaces
    • Section 70.8: Actions of group algebraic spaces
    • Section 70.9: Principal homogeneous spaces
    • Section 70.10: Equivariant quasi-coherent sheaves
    • Section 70.11: Groupoids in algebraic spaces
    • Section 70.12: Quasi-coherent sheaves on groupoids
    • Section 70.13: Crystals in quasi-coherent sheaves
    • Section 70.14: Groupoids and group spaces
    • Section 70.15: The stabilizer group algebraic space
    • Section 70.16: Restricting groupoids
    • Section 70.17: Invariant subspaces
    • Section 70.18: Quotient sheaves
    • Section 70.19: Quotient stacks
    • Section 70.20: Functoriality of quotient stacks
    • Section 70.21: The 2-cartesian square of a quotient stack
    • Section 70.22: The 2-coequalizer property of a quotient stack
    • Section 70.23: Explicit description of quotient stacks
    • Section 70.24: Restriction and quotient stacks
    • Section 70.25: Inertia and quotient stacks
    • Section 70.26: Gerbes and quotient stacks
    • Section 70.27: Quotient stacks and change of big site
    • Section 70.28: Separation conditions
  • Chapter 71: More on Groupoids in Spaces
    • Section 71.1: Introduction
    • Section 71.2: Notation
    • Section 71.3: Useful diagrams
    • Section 71.4: Local structure
    • Section 71.5: Groupoid of sections
    • Section 71.6: Properties of groupoids
    • Section 71.7: Comparing fibres
    • Section 71.8: Restricting groupoids
    • Section 71.9: Properties of groups over fields and groupoids on fields
    • Section 71.10: Group algebraic spaces over fields
    • Section 71.11: No rational curves on groups
    • Section 71.12: The finite part of a morphism
    • Section 71.13: Finite collections of arrows
    • Section 71.14: The finite part of a groupoid
    • Section 71.15: Étale localization of groupoid schemes
  • Chapter 72: Bootstrap
    • Section 72.1: Introduction
    • Section 72.2: Conventions
    • Section 72.3: Morphisms representable by algebraic spaces
    • Section 72.4: Properties of maps of presheaves representable by algebraic spaces
    • Section 72.5: Bootstrapping the diagonal
    • Section 72.6: Bootstrap
    • Section 72.7: Finding opens
    • Section 72.8: Slicing equivalence relations
    • Section 72.9: Quotient by a subgroupoid
    • Section 72.10: Final bootstrap
    • Section 72.11: Applications
    • Section 72.12: Algebraic spaces in the étale topology
  • Chapter 73: Pushouts of Algebraic Spaces
    • Section 73.1: Introduction
    • Section 73.2: Pushouts in the category of algebraic spaces
    • Section 73.3: Pushouts and derived categories
    • Section 73.4: Constructing elementary distinguished squares
    • Section 73.5: Formal glueing of quasi-coherent modules
    • Section 73.6: Formal glueing of algebraic spaces
    • Section 73.7: Coequalizers and glueing