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: Quasicoherent 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: Quasicompact 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 quasicoherent 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 nspace

Section 27.6: Vector bundles

Section 27.7: Cones

Section 27.8: Proj of a graded ring

Section 27.9: Quasicoherent 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: Quasicoherent 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: CohenMacaulay 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: Quasiaffine schemes

Section 28.19: Flat modules

Section 28.20: Locally free modules

Section 28.21: Locally projective modules

Section 28.22: Extending quasicoherent sheaves

Section 28.23: Gabber's result

Section 28.24: Sections with support in a closed subset

Section 28.25: Sections of quasicoherent sheaves

Section 28.26: Ample invertible sheaves

Section 28.27: Affine and quasiaffine schemes

Section 28.28: Quasicoherent 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 quasicoherent 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: Quasiaffine 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: Quasifinite 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: Quasiprojective 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: Zariski's Main Theorem (algebraic version)

Section 29.56: Universally bounded fibres

Chapter 30: Cohomology of Schemes

Section 30.1: Introduction

Section 30.2: Čech cohomology of quasicoherent sheaves

Section 30.3: Vanishing of cohomology

Section 30.4: Quasicoherence 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: Limits and dimensions of fibres

Section 32.17: Base change in top degree

Section 32.18: Glueing in closed fibres

Section 32.19: Application to modifications

Section 32.20: 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 CohenMacaulay 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: Euler characteristics

Section 33.33: Projective space

Section 33.34: Coherent sheaves on projective space

Section 33.35: Frobenii

Section 33.36: Glueing dimension one rings

Section 33.37: One dimensional Noetherian schemes

Section 33.38: The delta invariant

Section 33.39: The number of branches

Section 33.40: Normalization of one dimensional schemes

Section 33.41: Finding affine opens

Section 33.42: Curves

Section 33.43: Degrees on curves

Section 33.44: Numerical intersections

Section 33.45: Embedding dimension

Section 33.46: Bertini theorems

Section 33.47: EnriquesSeveriZariski

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 quasicoherent sheaves

Section 35.3: Descent for modules

Section 35.4: Descent for universally injective morphisms

Section 35.5: Fpqc descent of quasicoherent sheaves

Section 35.6: Galois descent for quasicoherent sheaves

Section 35.7: Descent of finiteness properties of modules

Section 35.8: Quasicoherent sheaves and topologies

Section 35.9: Parasitic modules

Section 35.10: Fpqc coverings are universal effective epimorphisms

Section 35.11: Descent of finiteness and smoothness properties of morphisms

Section 35.12: Local properties of schemes

Section 35.13: Properties of schemes local in the fppf topology

Section 35.14: Properties of schemes local in the syntomic topology

Section 35.15: Properties of schemes local in the smooth topology

Section 35.16: Variants on descending properties

Section 35.17: Germs of schemes

Section 35.18: Local properties of germs

Section 35.19: Properties of morphisms local on the target

Section 35.20: Properties of morphisms local in the fpqc topology on the target

Section 35.21: Properties of morphisms local in the fppf topology on the target

Section 35.22: Application of fpqc descent of properties of morphisms

Section 35.23: Properties of morphisms local on the source

Section 35.24: Properties of morphisms local in the fpqc topology on the source

Section 35.25: Properties of morphisms local in the fppf topology on the source

Section 35.26: Properties of morphisms local in the syntomic topology on the source

Section 35.27: Properties of morphisms local in the smooth topology on the source

Section 35.28: Properties of morphisms local in the étale topology on the source

Section 35.29: Properties of morphisms étale local on sourceandtarget

Section 35.30: Properties of morphisms of germs local on sourceandtarget

Section 35.31: Descent data for schemes over schemes

Section 35.32: Fully faithfulness of the pullback functors

Section 35.33: Descending types of morphisms

Section 35.34: Descending affine morphisms

Section 35.35: Descending quasiaffine morphisms

Section 35.36: 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 quasicoherent modules

Section 36.4: Total direct image

Section 36.5: Affine morphisms

Section 36.6: Cohomology with support in a closed

Section 36.7: The coherator

Section 36.8: The coherator for Noetherian schemes

Section 36.9: Koszul complexes

Section 36.10: Pseudocoherent 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 pseudocoherent 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 pseudocoherent 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: Kgroups

Section 36.39: Determinants of complexes

Section 36.40: Detecting Boundedness

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: Normalization revisited

Section 37.18: Normal morphisms

Section 37.19: Regular morphisms

Section 37.20: CohenMacaulay morphisms

Section 37.21: Slicing CohenMacaulay morphisms

Section 37.22: Generic fibres

Section 37.23: Relative assassins

Section 37.24: Reduced fibres

Section 37.25: Irreducible components of fibres

Section 37.26: Connected components of fibres

Section 37.27: Connected components meeting a section

Section 37.28: Dimension of fibres

Section 37.29: Bertini theorems

Section 37.30: Theorem of the cube

Section 37.31: Limit arguments

Section 37.32: Étale neighbourhoods

Section 37.33: Étale neighbourhoods and branches

Section 37.34: Slicing smooth morphisms

Section 37.35: Étale neighbourhoods and Artin approximation

Section 37.36: Finite free locally dominates étale

Section 37.37: Étale localization of quasifinite morphisms

Section 37.38: Étale localization of integral morphisms

Section 37.39: Zariski's Main Theorem

Section 37.40: Applications of Zariski's Main Theorem, I

Section 37.41: Applications of Zariski's Main Theorem, II

Section 37.42: Application to morphisms with connected fibres

Section 37.43: Application to the structure of finite type morphisms

Section 37.44: Application to the fppf topology

Section 37.45: Quasiprojective schemes

Section 37.46: Projective schemes

Section 37.47: Proj and Spec

Section 37.48: Closed points in fibres

Section 37.49: Stein factorization

Section 37.50: Descending separated locally quasifinite morphisms

Section 37.51: Relative finite presentation

Section 37.52: Relative pseudocoherence

Section 37.53: Pseudocoherent morphisms

Section 37.54: Perfect morphisms

Section 37.55: Local complete intersection morphisms

Section 37.56: Exact sequences of differentials and conormal sheaves

Section 37.57: Weakly étale morphisms

Section 37.58: Reduced fibre theorem

Section 37.59: Indquasiaffine morphisms

Section 37.60: Pushouts in the category of schemes, II

Section 37.61: Relative morphisms

Section 37.62: Characterizing pseudocoherent complexes, III

Section 37.63: Descent finiteness properties of complexes

Section 37.64: Relatively perfect objects

Section 37.65: Contracting rational curves

Section 37.66: Affine stratifications

Section 37.67: Universally open morphisms

Section 37.68: Weightings

Section 37.69: More on weightings

Section 37.70: Weightings and affine stratification numbers

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 MittagLeffler 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 quasicoherent sheaves

Section 39.13: Groupoids

Section 39.14: Quasicoherent sheaves on groupoids

Section 39.15: Colimits of quasicoherent 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 quasiprojective 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: CohenMacaulay 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 indquasiaffine 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