\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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\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*}
3 Topics in Scheme Theory

Chapter 41: Chow Homology and Chern Classes

Section 41.1: Introduction

Section 41.2: Periodic complexes and Herbrand quotients

Section 41.3: Calculuation of some multiplicities

Section 41.4: Preparation for tame symbols

Section 41.5: Tame symbols

Section 41.6: A key lemma

Section 41.7: Setup

Section 41.8: Cycles

Section 41.9: Cycle associated to a closed subscheme

Section 41.10: Cycle associated to a coherent sheaf

Section 41.11: Preparation for proper pushforward

Section 41.12: Proper pushforward

Section 41.13: Preparation for flat pullback

Section 41.14: Flat pullback

Section 41.15: Push and pull

Section 41.16: Preparation for principal divisors

Section 41.17: Principal divisors

Section 41.18: Principal divisors and pushforward

Section 41.19: Rational equivalence

Section 41.20: Rational equivalence and push and pull

Section 41.21: Rational equivalence and the projective line

Section 41.22: The divisor associated to an invertible sheaf

Section 41.23: Intersecting with an invertible sheaf

Section 41.24: Intersecting with an invertible sheaf and push and pull

Section 41.25: The key formula

Section 41.26: Intersecting with an invertible sheaf and rational equivalence

Section 41.27: Intersecting with effective Cartier divisors

Section 41.28: Gysin homomorphisms

Section 41.29: Relative effective Cartier divisors

Section 41.30: Affine bundles

Section 41.31: Bivariant intersection theory

Section 41.32: Projective space bundle formula

Section 41.33: The Chern classes of a vector bundle

Section 41.34: Intersecting with chern classes

Section 41.35: Polynomial relations among chern classes

Section 41.36: Additivity of chern classes

Section 41.37: The splitting principle

Section 41.38: The Chern character and tensor products

Section 41.39: Chern classes and the derived category

Section 41.40: Todd classes

Section 41.41: Degrees of zero cycles

Section 41.42: GrothendieckRiemannRoch

Section 41.43: Appendix A: Alternative approach to key lemma

Section 41.44: Appendix B: Alternative approaches

Chapter 42: Intersection Theory

Section 42.1: Introduction

Section 42.2: Conventions

Section 42.3: Cycles

Section 42.4: Cycle associated to closed subscheme

Section 42.5: Cycle associated to a coherent sheaf

Section 42.6: Proper pushforward

Section 42.7: Flat pullback

Section 42.8: Rational Equivalence

Section 42.9: Rational equivalence and rational functions

Section 42.10: Proper pushforward and rational equivalence

Section 42.11: Flat pullback and rational equivalence

Section 42.12: The short exact sequence for an open

Section 42.13: Proper intersections

Section 42.14: Intersection multiplicities using Tor formula

Section 42.15: Algebraic multiplicities

Section 42.16: Computing intersection multiplicities

Section 42.17: Intersection product using Tor formula

Section 42.18: Exterior product

Section 42.19: Reduction to the diagonal

Section 42.20: Associativity of intersections

Section 42.21: Flat pullback and intersection products

Section 42.22: Projection formula for flat proper morphisms

Section 42.23: Projections

Section 42.24: Moving Lemma

Section 42.25: Intersection products and rational equivalence

Section 42.26: Chow rings

Section 42.27: Pullback for a general morphism

Section 42.28: Pullback of cycles

Chapter 43: Picard Schemes of Curves

Section 43.1: Introduction

Section 43.2: Hilbert scheme of points

Section 43.3: Moduli of divisors on smooth curves

Section 43.4: The Picard functor

Section 43.5: A representability criterion

Section 43.6: The Picard scheme of a curve

Section 43.7: Some remarks on Picard groups

Chapter 44: Adequate Modules

Section 44.1: Introduction

Section 44.2: Conventions

Section 44.3: Adequate functors

Section 44.4: Higher exts of adequate functors

Section 44.5: Adequate modules

Section 44.6: Parasitic adequate modules

Section 44.7: Derived categories of adequate modules, I

Section 44.8: Pure extensions

Section 44.9: Higher exts of quasicoherent sheaves on the big site

Section 44.10: Derived categories of adequate modules, II

Chapter 45: Dualizing Complexes

Section 45.1: Introduction

Section 45.2: Essential surjections and injections

Section 45.3: Injective modules

Section 45.4: Projective covers

Section 45.5: Injective hulls

Section 45.6: Duality over Artinian local rings

Section 45.7: Injective hull of the residue field

Section 45.8: Deriving torsion

Section 45.9: Local cohomology

Section 45.10: Local cohomology for Noetherian rings

Section 45.11: Depth

Section 45.12: Torsion versus complete modules

Section 45.13: Trivial duality for a ring map

Section 45.14: Base change for trivial duality

Section 45.15: Dualizing complexes

Section 45.16: Dualizing complexes over local rings

Section 45.17: Dualizing complexes and dimension functions

Section 45.18: The local duality theorem

Section 45.19: Dualizing modules

Section 45.20: CohenMacaulay rings

Section 45.21: Gorenstein rings

Section 45.22: The ubiquity of dualizing complexes

Section 45.23: Formal fibres

Section 45.24: Upper shriek algebraically

Section 45.25: Relative dualizing complexes in the Noetherian case

Section 45.26: More on dualizing complexes

Section 45.27: Relative dualizing complexes

Chapter 46: Duality for Schemes

Section 46.1: Introduction

Section 46.2: Dualizing complexes on schemes

Section 46.3: Right adjoint of pushforward

Section 46.4: Right adjoint of pushforward and restriction to opens

Section 46.5: Right adjoint of pushforward and base change, I

Section 46.6: Right adjoint of pushforward and base change, II

Section 46.7: Right adjoint of pushforward and trace maps

Section 46.8: Right adjoint of pushforward and pullback

Section 46.9: Right adjoint of pushforward for closed immersions

Section 46.10: Right adjoint of pushforward for closed immersions and base change

Section 46.11: Right adjoint of pushforward for finite morphisms

Section 46.12: Right adjoint of pushforward for proper flat morphisms

Section 46.13: Right adjoint of pushforward for perfect proper morphisms

Section 46.14: Right adjoint of pushforward for effective Cartier divisors

Section 46.15: Right adjoint of pushforward in examples

Section 46.16: Compactifications

Section 46.17: Upper shriek functors

Section 46.18: Properties of upper shriek functors

Section 46.19: Base change for upper shriek

Section 46.20: A duality theory

Section 46.21: Glueing dualizing complexes

Section 46.22: Dimension functions

Section 46.23: Dualizing modules

Section 46.24: CohenMacaulay schemes

Section 46.25: Gorenstein schemes

Section 46.26: Gorenstein morphisms

Section 46.27: More on dualizing complexes

Section 46.28: Relative dualizing complexes

Section 46.29: The fundamental class of an lci morphism

Chapter 47: Discriminants and Differents

Section 47.1: Introduction

Section 47.2: Dualizing modules for quasifinite ring maps

Section 47.3: Discriminant of a finite locally free morphism

Section 47.4: Traces for flat quasifinite ring maps

Section 47.5: The Noether different

Section 47.6: The Kähler different

Section 47.7: The Dedekind different

Section 47.8: The different

Section 47.9: Quasifinite syntomic morphisms

Section 47.10: A formula for the different

Section 47.11: A generalization of the different

Section 47.12: Comparison with duality theory

Section 47.13: Quasifinite Gorenstein morphisms

Chapter 48: Local Cohomology

Section 48.1: Introduction

Section 48.2: Generalities

Section 48.3: Cohomological dimension

Section 48.4: More general supports

Section 48.5: Filtrations on local cohomology

Section 48.6: Finiteness of local cohomology, I

Section 48.7: Finiteness of pushforwards, I

Section 48.8: Depth and dimension

Section 48.9: Annihilators of local chomology, I

Section 48.10: Finiteness of local cohomology, II

Section 48.11: Finiteness of pushforwards, II

Section 48.12: Annihilators of local chomology, II

Section 48.13: Finiteness of local cohomology, III

Section 48.14: Improving coherent modules

Section 48.15: HartshorneLichtenbaum vanishing

Section 48.16: Frobenius action

Section 48.17: Structure of certain modules

Section 48.18: Additional structure on local cohomology

Chapter 49: Algebraic and Formal Geometry

Section 49.1: Introduction

Section 49.2: Formal sections, I

Section 49.3: Formal sections, II

Section 49.4: Formal sections, III

Section 49.5: MittagLeffler conditions

Section 49.6: Derived completion on a ringed site

Section 49.7: The theorem on formal functions

Section 49.8: Algebraization of local cohomology, I

Section 49.9: Algebraization of local cohomology, II

Section 49.10: Algebraization of local cohomology, III

Section 49.11: Algebraization of formal sections, I

Section 49.12: Algebraization of formal sections, II

Section 49.13: Algebraization of formal sections, III

Section 49.14: Application to connectedness

Section 49.15: The completion functor

Section 49.16: Algebraization of coherent formal modules, I

Section 49.17: Algebraization of coherent formal modules, II

Section 49.18: A distance function

Section 49.19: Algebraization of coherent formal modules, III

Section 49.20: Algebraization of coherent formal modules, IV

Section 49.21: Improving coherent formal modules

Section 49.22: Algebraization of coherent formal modules, V

Section 49.23: Algebraization of coherent formal modules, VI

Section 49.24: Application to the completion functor

Section 49.25: Application to Lefschetz theorems

Chapter 50: Algebraic Curves

Section 50.1: Introduction

Section 50.2: Curves and function fields

Section 50.3: Linear series

Section 50.4: Duality

Section 50.5: RiemannRoch

Section 50.6: Some vanishing results

Section 50.7: Very ample invertible sheaves

Section 50.8: The genus of a curve

Section 50.9: Plane curves

Section 50.10: Curves of genus zero

Section 50.11: Geometric genus

Section 50.12: RiemannHurwitz

Section 50.13: Inseparable maps

Section 50.14: Pushouts

Section 50.15: Glueing and squishing

Section 50.16: Multicross and nodal singularities

Section 50.17: Torsion in the Picard group

Section 50.18: Genus versus geometric genus

Section 50.19: Nodal curves

Section 50.20: Families of nodal curves

Section 50.21: More vanishing results

Section 50.22: Contracting rational tails

Section 50.23: Contracting rational bridges

Section 50.24: Contracting to a stable curve

Section 50.25: Vector fields

Chapter 51: Resolution of Surfaces

Section 51.1: Introduction

Section 51.2: A trace map in positive characteristic

Section 51.3: Quadratic transformations

Section 51.4: Dominating by quadratic transformations

Section 51.5: Dominating by normalized blowups

Section 51.6: Modifying over local rings

Section 51.7: Vanishing

Section 51.8: Boundedness

Section 51.9: Rational singularities

Section 51.10: Formal arcs

Section 51.11: Base change to the completion

Section 51.12: Rational double points

Section 51.13: Implied properties

Section 51.14: Resolution

Section 51.15: Embedded resolution

Section 51.16: Contracting exceptional curves

Section 51.17: Factorization birational maps

Chapter 52: Semistable Reduction

Section 52.1: Introduction

Section 52.2: Linear algebra

Section 52.3: Numerical types

Section 52.4: The Picard group of a numerical type

Section 52.5: Classification of proper subgraphs

Section 52.6: Classification of minimal type for genus zero and one

Section 52.7: Bounding invariants of numerical types

Section 52.8: Models

Section 52.9: The geometry of a regular model

Section 52.10: Uniqueness of the minimal model

Section 52.11: A formula for the genus

Section 52.12: Blowing down exceptional curves

Section 52.13: Picard groups of models

Section 52.14: Semistable reduction

Section 52.15: Semistable reduction in genus zero

Section 52.16: Semistable reduction in genus one

Section 52.17: Semistable reduction in genus at least two

Section 52.18: Semistable reduction for curves

Chapter 53: Fundamental Groups of Schemes

Section 53.1: Introduction

Section 53.2: Schemes étale over a point

Section 53.3: Galois categories

Section 53.4: Functors and homomorphisms

Section 53.5: Finite étale morphisms

Section 53.6: Fundamental groups

Section 53.7: Galois covers of connected schemes

Section 53.8: Topological invariance of the fundamental group

Section 53.9: Finite étale covers of proper schemes

Section 53.10: Local connectedness

Section 53.11: Fundamental groups of normal schemes

Section 53.12: Group actions and integral closure

Section 53.13: Ramification theory

Section 53.14: Geometric and arithmetic fundamental groups

Section 53.15: Homotopy exact sequence

Section 53.16: Specialization maps

Section 53.17: Restriction to a closed subscheme

Section 53.18: Pushouts and fundamental groups

Section 53.19: Finite étale covers of punctured spectra, I

Section 53.20: Purity in local case, I

Section 53.21: Purity of branch locus

Section 53.22: Finite étale covers of punctured spectra, II

Section 53.23: Finite étale covers of punctured spectra, III

Section 53.24: Finite étale covers of punctured spectra, IV

Section 53.25: Purity in local case, II

Section 53.26: Purity in local case, III

Section 53.27: Lefschetz for the fundamental group

Section 53.28: Purity of ramification locus

Section 53.29: Affineness of complement of ramification locus

Section 53.30: Specialization maps in the smooth proper case

Section 53.31: Tame ramification

Chapter 54: Étale Cohomology

Section 54.1: Introduction

Section 54.2: Which sections to skip on a first reading?

Section 54.3: Prologue

Section 54.4: The étale topology

Section 54.5: Feats of the étale topology

Section 54.6: A computation

Section 54.7: Nontorsion coefficients

Section 54.8: Sheaf theory

Section 54.9: Presheaves

Section 54.10: Sites

Section 54.11: Sheaves

Section 54.12: The example of Gsets

Section 54.13: Sheafification

Section 54.14: Cohomology

Section 54.15: The fpqc topology

Section 54.16: Faithfully flat descent

Section 54.17: Quasicoherent sheaves

Section 54.18: Čech cohomology

Section 54.19: The Čechtocohomology spectral sequence

Section 54.20: Big and small sites of schemes

Section 54.21: The étale topos

Section 54.22: Cohomology of quasicoherent sheaves

Section 54.23: Examples of sheaves

Section 54.24: Picard groups

Section 54.25: The étale site

Section 54.26: Étale morphisms

Section 54.27: Étale coverings

Section 54.28: Kummer theory

Section 54.29: Neighborhoods, stalks and points

Section 54.30: Points in other topologies

Section 54.31: Supports of abelian sheaves

Section 54.32: Henselian rings

Section 54.33: Stalks of the structure sheaf

Section 54.34: Functoriality of small étale topos

Section 54.35: Direct images

Section 54.36: Inverse image

Section 54.37: Functoriality of big topoi

Section 54.38: Functoriality and sheaves of modules

Section 54.39: Comparing topologies

Section 54.40: Recovering morphisms

Section 54.41: Push and pull

Section 54.42: Property (A)

Section 54.43: Property (B)

Section 54.44: Property (C)

Section 54.45: Topological invariance of the small étale site

Section 54.46: Closed immersions and pushforward

Section 54.47: Integral universally injective morphisms

Section 54.48: Big sites and pushforward

Section 54.49: Exactness of big lower shriek

Section 54.50: Étale cohomology

Section 54.51: Colimits

Section 54.52: Stalks of higher direct images

Section 54.53: The Leray spectral sequence

Section 54.54: Vanishing of finite higher direct images

Section 54.55: Galois action on stalks

Section 54.56: Group cohomology

Section 54.57: Continuous group cohomology

Section 54.58: Cohomology of a point

Section 54.59: Cohomology of curves

Section 54.60: Brauer groups

Section 54.61: The Brauer group of a scheme

Section 54.62: The ArtinSchreier sequence

Section 54.63: Locally constant sheaves

Section 54.64: Locally constant sheaves and the fundamental group

Section 54.65: Méthode de la trace

Section 54.66: Galois cohomology

Section 54.67: Higher vanishing for the multiplicative group

Section 54.68: Picard groups of curves

Section 54.69: Extension by zero

Section 54.70: Constructible sheaves

Section 54.71: Auxiliary lemmas on morphisms

Section 54.72: More on constructible sheaves

Section 54.73: Constructible sheaves on Noetherian schemes

Section 54.74: Torsion sheaves

Section 54.75: Cohomology with support in a closed subscheme

Section 54.76: Schemes with strictly henselian local rings

Section 54.77: Affine analog of proper base change

Section 54.78: Cohomology of torsion sheaves on curves

Section 54.79: First cohomology of proper schemes

Section 54.80: Preliminaries on base change

Section 54.81: Base change for pushforward

Section 54.82: Base change for higher direct images

Section 54.83: Smooth base change

Section 54.84: Applications of smooth base change

Section 54.85: The proper base change theorem

Section 54.86: Applications of proper base change

Section 54.87: Cohomological dimension

Section 54.88: Finite cohomological dimension

Section 54.89: Künneth in étale cohomology

Section 54.90: Comparing chaotic and Zariski topologies

Section 54.91: Comparing big and small topoi

Section 54.92: Comparing fppf and étale topologies

Section 54.93: Comparing fppf and étale topologies: modules

Section 54.94: Comparing ph and étale topologies

Section 54.95: Comparing h and étale topologies

Section 54.96: Blow up squares and étale cohomology

Section 54.97: Almost blow up squares and the h topology

Section 54.98: Cohomology of the structure sheaf in the h topology

Section 54.99: The trace formula

Section 54.100: Frobenii

Section 54.101: Traces

Section 54.102: Why derived categories?

Section 54.103: Derived categories

Section 54.104: Filtered derived category

Section 54.105: Filtered derived functors

Section 54.106: Application of filtered complexes

Section 54.107: Perfectness

Section 54.108: Filtrations and perfect complexes

Section 54.109: Characterizing perfect objects

Section 54.110: Complexes with constructible cohomology

Section 54.111: Cohomology of nice complexes

Section 54.112: Lefschetz numbers

Section 54.113: Preliminaries and sorites

Section 54.114: Proof of the trace formula

Section 54.115: Applications

Section 54.116: On ladic sheaves

Section 54.117: Lfunctions

Section 54.118: Cohomological interpretation

Section 54.119: List of things which we should add above

Section 54.120: Examples of Lfunctions

Section 54.121: Constant sheaves

Section 54.122: The Legendre family

Section 54.123: Exponential sums

Section 54.124: Trace formula in terms of fundamental groups

Section 54.125: Fundamental groups

Section 54.126: Profinite groups, cohomology and homology

Section 54.127: Cohomology of curves, revisited

Section 54.128: Abstract trace formula

Section 54.129: Automorphic forms and sheaves

Section 54.130: Counting points

Section 54.131: Precise form of Chebotarev

Section 54.132: How many primes decompose completely?

Section 54.133: How many points are there really?

Chapter 55: Crystalline Cohomology

Section 55.1: Introduction

Section 55.2: Divided power envelope

Section 55.3: Some explicit divided power thickenings

Section 55.4: Compatibility

Section 55.5: Affine crystalline site

Section 55.6: Module of differentials

Section 55.7: Divided power schemes

Section 55.8: The big crystalline site

Section 55.9: The crystalline site

Section 55.10: Sheaves on the crystalline site

Section 55.11: Crystals in modules

Section 55.12: Sheaf of differentials

Section 55.13: Two universal thickenings

Section 55.14: The de Rham complex

Section 55.15: Connections

Section 55.16: Cosimplicial algebra

Section 55.17: Crystals in quasicoherent modules

Section 55.18: General remarks on cohomology

Section 55.19: Cosimplicial preparations

Section 55.20: Divided power Poincaré lemma

Section 55.21: Cohomology in the affine case

Section 55.22: Two counter examples

Section 55.23: Applications

Section 55.24: Some further results

Section 55.25: Pulling back along purely inseparable maps

Section 55.26: Frobenius action on crystalline cohomology

Chapter 56: Proétale Cohomology

Section 56.1: Introduction

Section 56.2: Some topology

Section 56.3: Local isomorphisms

Section 56.4: IndZariski algebra

Section 56.5: Constructing wlocal affine schemes

Section 56.6: Identifying local rings versus indZariski

Section 56.7: Indétale algebra

Section 56.8: Constructing indétale algebras

Section 56.9: Weakly étale versus proétale

Section 56.10: The V topology and the proh topology

Section 56.11: Constructing wcontractible covers

Section 56.12: The proétale site

Section 56.13: Points of the proétale site

Section 56.14: Compact generation

Section 56.15: Derived completion in the constant Noetherian case

Section 56.16: Derived completion on the proétale site

Section 56.17: Comparison with the étale site

Section 56.18: Cohomology of a point

Section 56.19: Weakly contractible hypercoverings

Section 56.20: Functoriality of the proétale site

Section 56.21: Finite morphisms and proétale sites

Section 56.22: Closed immersions and proétale sites

Section 56.23: Extension by zero

Section 56.24: Constructible sheaves on the proétale site

Section 56.25: Constructible adic sheaves

Section 56.26: A suitable derived category

Section 56.27: Proper base change