4 Algebraic Spaces

Chapter 62: Algebraic Spaces

Section 62.1: Introduction

Section 62.2: General remarks

Section 62.3: Representable morphisms of presheaves

Section 62.4: Lists of useful properties of morphisms of schemes

Section 62.5: Properties of representable morphisms of presheaves

Section 62.6: Algebraic spaces

Section 62.7: Fibre products of algebraic spaces

Section 62.8: Glueing algebraic spaces

Section 62.9: Presentations of algebraic spaces

Section 62.10: Algebraic spaces and equivalence relations

Section 62.11: Algebraic spaces, retrofitted

Section 62.12: Immersions and Zariski coverings of algebraic spaces

Section 62.13: Separation conditions on algebraic spaces

Section 62.14: Examples of algebraic spaces

Section 62.15: Change of big site

Section 62.16: Change of base scheme

Chapter 63: Properties of Algebraic Spaces

Section 63.1: Introduction

Section 63.2: Conventions

Section 63.3: Separation axioms

Section 63.4: Points of algebraic spaces

Section 63.5: Quasicompact spaces

Section 63.6: Special coverings

Section 63.7: Properties of Spaces defined by properties of schemes

Section 63.8: Constructible sets

Section 63.9: Dimension at a point

Section 63.10: Dimension of local rings

Section 63.11: Generic points

Section 63.12: Reduced spaces

Section 63.13: The schematic locus

Section 63.14: Obtaining a scheme

Section 63.15: Points on quasiseparated spaces

Section 63.16: Étale morphisms of algebraic spaces

Section 63.17: Spaces and fpqc coverings

Section 63.18: The étale site of an algebraic space

Section 63.19: Points of the small étale site

Section 63.20: Supports of abelian sheaves

Section 63.21: The structure sheaf of an algebraic space

Section 63.22: Stalks of the structure sheaf

Section 63.23: Local irreducibility

Section 63.24: Noetherian spaces

Section 63.25: Regular algebraic spaces

Section 63.26: Sheaves of modules on algebraic spaces

Section 63.27: Étale localization

Section 63.28: Recovering morphisms

Section 63.29: Quasicoherent sheaves on algebraic spaces

Section 63.30: Properties of modules

Section 63.31: Locally projective modules

Section 63.32: Quasicoherent sheaves and presentations

Section 63.33: Morphisms towards schemes

Section 63.34: Quotients by free actions

Chapter 64: Morphisms of Algebraic Spaces

Section 64.1: Introduction

Section 64.2: Conventions

Section 64.3: Properties of representable morphisms

Section 64.4: Separation axioms

Section 64.5: Surjective morphisms

Section 64.6: Open morphisms

Section 64.7: Submersive morphisms

Section 64.8: Quasicompact morphisms

Section 64.9: Universally closed morphisms

Section 64.10: Monomorphisms

Section 64.11: Pushforward of quasicoherent sheaves

Section 64.12: Immersions

Section 64.13: Closed immersions

Section 64.14: Closed immersions and quasicoherent sheaves

Section 64.15: Supports of modules

Section 64.16: Scheme theoretic image

Section 64.17: Scheme theoretic closure and density

Section 64.18: Dominant morphisms

Section 64.19: Universally injective morphisms

Section 64.20: Affine morphisms

Section 64.21: Quasiaffine morphisms

Section 64.22: Types of morphisms étale local on sourceandtarget

Section 64.23: Morphisms of finite type

Section 64.24: Points and geometric points

Section 64.25: Points of finite type

Section 64.26: Nagata spaces

Section 64.27: Quasifinite morphisms

Section 64.28: Morphisms of finite presentation

Section 64.29: Constructible sets

Section 64.30: Flat morphisms

Section 64.31: Flat modules

Section 64.32: Generic flatness

Section 64.33: Relative dimension

Section 64.34: Morphisms and dimensions of fibres

Section 64.35: The dimension formula

Section 64.36: Syntomic morphisms

Section 64.37: Smooth morphisms

Section 64.38: Unramified morphisms

Section 64.39: Étale morphisms

Section 64.40: Proper morphisms

Section 64.41: Valuative criteria

Section 64.42: Valuative criterion for universal closedness

Section 64.43: Valuative criterion of separatedness

Section 64.44: Valuative criterion of properness

Section 64.45: Integral and finite morphisms

Section 64.46: Finite locally free morphisms

Section 64.47: Rational maps

Section 64.48: Relative normalization of algebraic spaces

Section 64.49: Normalization

Section 64.50: Separated, locally quasifinite morphisms

Section 64.51: Applications

Section 64.52: Zariski's Main Theorem (representable case)

Section 64.53: Universal homeomorphisms

Chapter 65: Decent Algebraic Spaces

Section 65.1: Introduction

Section 65.2: Conventions

Section 65.3: Universally bounded fibres

Section 65.4: Finiteness conditions and points

Section 65.5: Conditions on algebraic spaces

Section 65.6: Reasonable and decent algebraic spaces

Section 65.7: Points and specializations

Section 65.8: Stratifying algebraic spaces by schemes

Section 65.9: Integral cover by a scheme

Section 65.10: Schematic locus

Section 65.11: Residue fields and henselian local rings

Section 65.12: Points on decent spaces

Section 65.13: Reduced singleton spaces

Section 65.14: Decent spaces

Section 65.15: Locally separated spaces

Section 65.16: Valuative criterion

Section 65.17: Relative conditions

Section 65.18: Points of fibres

Section 65.19: Monomorphisms

Section 65.20: Generic points

Section 65.21: Generically finite morphisms

Section 65.22: Birational morphisms

Section 65.23: Jacobson spaces

Section 65.24: Local irreducibility

Section 65.25: Catenary algebraic spaces

Chapter 66: Cohomology of Algebraic Spaces

Section 66.1: Introduction

Section 66.2: Conventions

Section 66.3: Higher direct images

Section 66.4: Finite morphisms

Section 66.5: Colimits and cohomology

Section 66.6: The alternating Čech complex

Section 66.7: Higher vanishing for quasicoherent sheaves

Section 66.8: Vanishing for higher direct images

Section 66.9: Cohomology with support in a closed subspace

Section 66.10: Vanishing above the dimension

Section 66.11: Cohomology and base change, I

Section 66.12: Coherent modules on locally Noetherian algebraic spaces

Section 66.13: Coherent sheaves on Noetherian spaces

Section 66.14: Devissage of coherent sheaves

Section 66.15: Limits of coherent modules

Section 66.16: Vanishing of cohomology

Section 66.17: Finite morphisms and affines

Section 66.18: A weak version of Chow's lemma

Section 66.19: Noetherian valuative criterion

Section 66.20: Higher direct images of coherent sheaves

Section 66.21: The theorem on formal functions

Section 66.22: Applications of the theorem on formal functions

Chapter 67: Limits of Algebraic Spaces

Section 67.1: Introduction

Section 67.2: Conventions

Section 67.3: Morphisms of finite presentation

Section 67.4: Limits of algebraic spaces

Section 67.5: Descending properties

Section 67.6: Descending properties of morphisms

Section 67.7: Descending relative objects

Section 67.8: Absolute Noetherian approximation

Section 67.9: Applications

Section 67.10: Relative approximation

Section 67.11: Finite type closed in finite presentation

Section 67.12: Approximating proper morphisms

Section 67.13: Embedding into affine space

Section 67.14: Sections with support in a closed subset

Section 67.15: Characterizing affine spaces

Section 67.16: Finite cover by a scheme

Section 67.17: Obtaining schemes

Section 67.18: Glueing in closed fibres

Section 67.19: Application to modifications

Section 67.20: Universally closed morphisms

Section 67.21: Noetherian valuative criterion

Section 67.22: Descending finite type spaces

Chapter 68: Divisors on Algebraic Spaces

Section 68.1: Introduction

Section 68.2: Associated and weakly associated points

Section 68.3: Morphisms and weakly associated points

Section 68.4: Relative weak assassin

Section 68.5: Fitting ideals

Section 68.6: Effective Cartier divisors

Section 68.7: Effective Cartier divisors and invertible sheaves

Section 68.8: Effective Cartier divisors on Noetherian spaces

Section 68.9: Relative effective Cartier divisors

Section 68.10: Meromorphic functions and sections

Section 68.11: Relative Proj

Section 68.12: Functoriality of relative proj

Section 68.13: Invertible sheaves and morphisms into relative Proj

Section 68.14: Relatively ample sheaves

Section 68.15: Relative ampleness and cohomology

Section 68.16: Closed subspaces of relative proj

Section 68.17: Blowing up

Section 68.18: Strict transform

Section 68.19: Admissible blowups

Chapter 69: Algebraic Spaces over Fields

Section 69.1: Introduction

Section 69.2: Conventions

Section 69.3: Generically finite morphisms

Section 69.4: Integral algebraic spaces

Section 69.5: Morphisms between integral algebraic spaces

Section 69.6: Weil divisors

Section 69.7: The Weil divisor class associated to an invertible module

Section 69.8: Modifications and alterations

Section 69.9: Schematic locus

Section 69.10: Schematic locus and field extension

Section 69.11: Geometrically reduced algebraic spaces

Section 69.12: Geometrically connected algebraic spaces

Section 69.13: Geometrically irreducible algebraic spaces

Section 69.14: Geometrically integral algebraic spaces

Section 69.15: Dimension

Section 69.16: Spaces smooth over fields

Section 69.17: Euler characteristics

Section 69.18: Numerical intersections

Chapter 70: Topologies on Algebraic Spaces

Section 70.1: Introduction

Section 70.2: The general procedure

Section 70.3: Zariski topology

Section 70.4: Étale topology

Section 70.5: Smooth topology

Section 70.6: Syntomic topology

Section 70.7: Fppf topology

Section 70.8: The ph topology

Section 70.9: Fpqc topology

Chapter 71: Descent and Algebraic Spaces

Section 71.1: Introduction

Section 71.2: Conventions

Section 71.3: Descent data for quasicoherent sheaves

Section 71.4: Fpqc descent of quasicoherent sheaves

Section 71.5: Descent of finiteness properties of modules

Section 71.6: Fpqc coverings

Section 71.7: Descent of finiteness and smoothness properties of morphisms

Section 71.8: Descending properties of spaces

Section 71.9: Descending properties of morphisms

Section 71.10: Descending properties of morphisms in the fpqc topology

Section 71.11: Descending properties of morphisms in the fppf topology

Section 71.12: Application of descent of properties of morphisms

Section 71.13: Properties of morphisms local on the source

Section 71.14: Properties of morphisms local in the fpqc topology on the source

Section 71.15: Properties of morphisms local in the fppf topology on the source

Section 71.16: Properties of morphisms local in the syntomic topology on the source

Section 71.17: Properties of morphisms local in the smooth topology on the source

Section 71.18: Properties of morphisms local in the étale topology on the source

Section 71.19: Properties of morphisms smooth local on sourceandtarget

Section 71.20: Properties of morphisms étalesmooth local on sourceandtarget

Section 71.21: Descent data for spaces over spaces

Section 71.22: Descent data in terms of sheaves

Chapter 72: Derived Categories of Spaces

Section 72.1: Introduction

Section 72.2: Conventions

Section 72.3: Generalities

Section 72.4: Derived category of quasicoherent modules on the small étale site

Section 72.5: Derived category of quasicoherent modules

Section 72.6: Total direct image

Section 72.7: Being proper over a base

Section 72.8: Derived category of coherent modules

Section 72.9: Induction principle

Section 72.10: MayerVietoris

Section 72.11: The coherator

Section 72.12: The coherator for Noetherian spaces

Section 72.13: Pseudocoherent and perfect complexes

Section 72.14: Approximation by perfect complexes

Section 72.15: Generating derived categories

Section 72.16: Compact and perfect objects

Section 72.17: Derived categories as module categories

Section 72.18: Characterizing pseudocoherent complexes, I

Section 72.19: The coherator revisited

Section 72.20: Cohomology and base change, IV

Section 72.21: Cohomology and base change, V

Section 72.22: Producing perfect complexes

Section 72.23: A projection formula for Ext

Section 72.24: Limits and derived categories

Section 72.25: Cohomology and base change, VI

Section 72.26: Perfect complexes

Section 72.27: Other applications

Chapter 73: More on Morphisms of Spaces

Section 73.1: Introduction

Section 73.2: Conventions

Section 73.3: Radicial morphisms

Section 73.4: Monomorphisms

Section 73.5: Conormal sheaf of an immersion

Section 73.6: The normal cone of an immersion

Section 73.7: Sheaf of differentials of a morphism

Section 73.8: Topological invariance of the étale site

Section 73.9: Thickenings

Section 73.10: Morphisms of thickenings

Section 73.11: Picard groups of thickenings

Section 73.12: First order infinitesimal neighbourhood

Section 73.13: Formally smooth, étale, unramified transformations

Section 73.14: Formally unramified morphisms

Section 73.15: Universal first order thickenings

Section 73.16: Formally étale morphisms

Section 73.17: Infinitesimal deformations of maps

Section 73.18: Infinitesimal deformations of algebraic spaces

Section 73.19: Formally smooth morphisms

Section 73.20: Smoothness over a Noetherian base

Section 73.21: The naive cotangent complex

Section 73.22: Openness of the flat locus

Section 73.23: Critère de platitude par fibres

Section 73.24: Flatness over a Noetherian base

Section 73.25: Normalization revisited

Section 73.26: CohenMacaulay morphisms

Section 73.27: Gorenstein morphisms

Section 73.28: Slicing CohenMacaulay morphisms

Section 73.29: Reduced fibres

Section 73.30: Connected components of fibres

Section 73.31: Dimension of fibres

Section 73.32: Catenary algebraic spaces

Section 73.33: Étale localization of morphisms

Section 73.34: Zariski's Main Theorem

Section 73.35: Applications of Zariski's Main Theorem, I

Section 73.36: Stein factorization

Section 73.37: Extending properties from an open

Section 73.38: Blowing up and flatness

Section 73.39: Applications

Section 73.40: Chow's lemma

Section 73.41: Variants of Chow's Lemma

Section 73.42: Grothendieck's existence theorem

Section 73.43: Grothendieck's algebraization theorem

Section 73.44: Regular immersions

Section 73.45: Relative pseudocoherence

Section 73.46: Pseudocoherent morphisms

Section 73.47: Perfect morphisms

Section 73.48: Local complete intersection morphisms

Section 73.49: When is a morphism an isomorphism?

Section 73.50: Exact sequences of differentials and conormal sheaves

Section 73.51: Characterizing pseudocoherent complexes, II

Section 73.52: Relatively perfect objects

Section 73.53: Theorem of the cube

Section 73.54: Descent of finiteness properties of complexes

Section 73.55: Families of nodal curves

Chapter 74: Flatness on Algebraic Spaces

Section 74.1: Introduction

Section 74.2: Impurities

Section 74.3: Relatively pure modules

Section 74.4: Flat finite type modules

Section 74.5: Flat finitely presented modules

Section 74.6: A criterion for purity

Section 74.7: Flattening functors

Section 74.8: Making a map zero

Section 74.9: Flattening a map

Section 74.10: Flattening in the local case

Section 74.11: Universal flattening

Section 74.12: Grothendieck's Existence Theorem

Section 74.13: Grothendieck's Existence Theorem, bis

Chapter 75: Groupoids in Algebraic Spaces

Section 75.1: Introduction

Section 75.2: Conventions

Section 75.3: Notation

Section 75.4: Equivalence relations

Section 75.5: Group algebraic spaces

Section 75.6: Properties of group algebraic spaces

Section 75.7: Examples of group algebraic spaces

Section 75.8: Actions of group algebraic spaces

Section 75.9: Principal homogeneous spaces

Section 75.10: Equivariant quasicoherent sheaves

Section 75.11: Groupoids in algebraic spaces

Section 75.12: Quasicoherent sheaves on groupoids

Section 75.13: Crystals in quasicoherent sheaves

Section 75.14: Groupoids and group spaces

Section 75.15: The stabilizer group algebraic space

Section 75.16: Restricting groupoids

Section 75.17: Invariant subspaces

Section 75.18: Quotient sheaves

Section 75.19: Quotient stacks

Section 75.20: Functoriality of quotient stacks

Section 75.21: The 2cartesian square of a quotient stack

Section 75.22: The 2coequalizer property of a quotient stack

Section 75.23: Explicit description of quotient stacks

Section 75.24: Restriction and quotient stacks

Section 75.25: Inertia and quotient stacks

Section 75.26: Gerbes and quotient stacks

Section 75.27: Quotient stacks and change of big site

Section 75.28: Separation conditions

Chapter 76: More on Groupoids in Spaces

Section 76.1: Introduction

Section 76.2: Notation

Section 76.3: Useful diagrams

Section 76.4: Local structure

Section 76.5: Groupoid of sections

Section 76.6: Properties of groupoids

Section 76.7: Comparing fibres

Section 76.8: Restricting groupoids

Section 76.9: Properties of groups over fields and groupoids on fields

Section 76.10: Group algebraic spaces over fields

Section 76.11: No rational curves on groups

Section 76.12: The finite part of a morphism

Section 76.13: Finite collections of arrows

Section 76.14: The finite part of a groupoid

Section 76.15: Étale localization of groupoid schemes

Chapter 77: Bootstrap

Section 77.1: Introduction

Section 77.2: Conventions

Section 77.3: Morphisms representable by algebraic spaces

Section 77.4: Properties of maps of presheaves representable by algebraic spaces

Section 77.5: Bootstrapping the diagonal

Section 77.6: Bootstrap

Section 77.7: Finding opens

Section 77.8: Slicing equivalence relations

Section 77.9: Quotient by a subgroupoid

Section 77.10: Final bootstrap

Section 77.11: Applications

Section 77.12: Algebraic spaces in the étale topology

Chapter 78: Pushouts of Algebraic Spaces

Section 78.1: Introduction

Section 78.2: Pushouts in the category of algebraic spaces

Section 78.3: Pushouts and derived categories

Section 78.4: Constructing elementary distinguished squares

Section 78.5: Formal glueing of quasicoherent modules

Section 78.6: Formal glueing of algebraic spaces

Section 78.7: Glueing and the BeauvilleLaszlo theorem

Section 78.8: Coequalizers and glueing

Section 78.9: Compactifications