\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: Quasicompact 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 quasiseparated 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: Quasicoherent sheaves on algebraic spaces

Section 58.30: Properties of modules

Section 58.31: Locally projective modules

Section 58.32: Quasicoherent 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: Quasicompact morphisms

Section 59.9: Universally closed morphisms

Section 59.10: Monomorphisms

Section 59.11: Pushforward of quasicoherent sheaves

Section 59.12: Immersions

Section 59.13: Closed immersions

Section 59.14: Closed immersions and quasicoherent 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: Quasiaffine morphisms

Section 59.22: Types of morphisms étale local on sourceandtarget

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: Quasifinite 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 quasifinite 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 quasicoherent 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 quasicoherent sheaves

Section 66.4: Fpqc descent of quasicoherent 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 sourceandtarget

Section 66.20: Properties of morphisms étalesmooth local on sourceandtarget

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 quasicoherent modules on the small étale site

Section 67.5: Derived category of quasicoherent 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: MayerVietoris

Section 67.11: The coherator

Section 67.12: The coherator for Noetherian spaces

Section 67.13: Pseudocoherent 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 pseudocoherent 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: CohenMacaulay morphisms

Section 68.27: Gorenstein morphisms

Section 68.28: Slicing CohenMacaulay 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 pseudocoherence

Section 68.45: Pseudocoherent 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 pseudocoherent 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 quasicoherent sheaves

Section 70.11: Groupoids in algebraic spaces

Section 70.12: Quasicoherent sheaves on groupoids

Section 70.13: Crystals in quasicoherent 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 2cartesian square of a quotient stack

Section 70.22: The 2coequalizer 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 quasicoherent modules

Section 73.6: Formal glueing of algebraic spaces

Section 73.7: Coequalizers and glueing