\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}
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\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: Quasicoherent 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: Quasicompact 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 quasicoherent 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 nspace

Section 26.6: Vector bundles

Section 26.7: Cones

Section 26.8: Proj of a graded ring

Section 26.9: Quasicoherent 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: Quasicoherent 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: CohenMacaulay 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: Quasiaffine schemes

Section 27.19: Flat modules

Section 27.20: Locally free modules

Section 27.21: Locally projective modules

Section 27.22: Extending quasicoherent sheaves

Section 27.23: Gabber's result

Section 27.24: Sections with support in a closed subset

Section 27.25: Sections of quasicoherent sheaves

Section 27.26: Ample invertible sheaves

Section 27.27: Affine and quasiaffine schemes

Section 27.28: Quasicoherent 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 quasicoherent 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: Quasiaffine 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: Quasifinite 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: Quasiprojective 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 quasicoherent sheaves

Section 29.3: Vanishing of cohomology

Section 29.4: Quasicoherence 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 CohenMacaulay 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 quasicoherent sheaves

Section 34.3: Descent for modules

Section 34.4: Descent for universally injective morphisms

Section 34.5: Fpqc descent of quasicoherent sheaves

Section 34.6: Galois descent for quasicoherent sheaves

Section 34.7: Descent of finiteness properties of modules

Section 34.8: Quasicoherent 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 sourceandtarget

Section 34.30: Properties of morphisms of germs local on sourceandtarget

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 quasiaffine 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 quasicoherent 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: Pseudocoherent 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 pseudocoherent 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 pseudocoherent 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: CohenMacaulay morphisms

Section 36.21: Slicing CohenMacaulay 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 quasifinite 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: Quasiprojective 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 quasifinite morphisms

Section 36.48: Relative finite presentation

Section 36.49: Relative pseudocoherence

Section 36.50: Pseudocoherent 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: Indquasiaffine morphisms

Section 36.57: Pushouts in the category of schemes, II

Section 36.58: Relative morphisms

Section 36.59: Characterizing pseudocoherent 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 MittagLeffler 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 quasicoherent sheaves

Section 38.13: Groupoids

Section 38.14: Quasicoherent sheaves on groupoids

Section 38.15: Colimits of quasicoherent 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 quasiprojective 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: CohenMacaulay 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 indquasiaffine 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