3 Topics in Scheme Theory

Chapter 42: Chow Homology and Chern Classes

Section 42.1: Introduction

Section 42.2: Periodic complexes and Herbrand quotients

Section 42.3: Calculation of some multiplicities

Section 42.4: Preparation for tame symbols

Section 42.5: Tame symbols

Section 42.6: A key lemma

Section 42.7: Setup

Section 42.8: Cycles

Section 42.9: Cycle associated to a closed subscheme

Section 42.10: Cycle associated to a coherent sheaf

Section 42.11: Preparation for proper pushforward

Section 42.12: Proper pushforward

Section 42.13: Preparation for flat pullback

Section 42.14: Flat pullback

Section 42.15: Push and pull

Section 42.16: Preparation for principal divisors

Section 42.17: Principal divisors

Section 42.18: Principal divisors and pushforward

Section 42.19: Rational equivalence

Section 42.20: Rational equivalence and push and pull

Section 42.21: Rational equivalence and the projective line

Section 42.22: Chow groups and Kgroups

Section 42.23: The divisor associated to an invertible sheaf

Section 42.24: Intersecting with an invertible sheaf

Section 42.25: Intersecting with an invertible sheaf and push and pull

Section 42.26: The key formula

Section 42.27: Intersecting with an invertible sheaf and rational equivalence

Section 42.28: Gysin homomorphisms

Section 42.29: Gysin homomorphisms and rational equivalence

Section 42.30: Relative effective Cartier divisors

Section 42.31: Affine bundles

Section 42.32: Bivariant intersection theory

Section 42.33: Chow cohomology and the first chern class

Section 42.34: Lemmas on bivariant classes

Section 42.35: Projective space bundle formula

Section 42.36: The Chern classes of a vector bundle

Section 42.37: Intersecting with chern classes

Section 42.38: Polynomial relations among chern classes

Section 42.39: Additivity of chern classes

Section 42.40: Degrees of zero cycles

Section 42.41: Cycles of given codimension

Section 42.42: The splitting principle

Section 42.43: Chern classes and sections

Section 42.44: The Chern character and tensor products

Section 42.45: Chern classes and the derived category

Section 42.46: A baby case of localized chern classes

Section 42.47: Gysin at infinity

Section 42.48: Preparation for localized chern classes

Section 42.49: Localized chern classes

Section 42.50: Two technical lemmas

Section 42.51: Properties of localized chern classes

Section 42.52: Blowing up at infinity

Section 42.53: Higher codimension gysin homomorphisms

Section 42.54: Calculating some classes

Section 42.55: An Adams operator

Section 42.56: Chow groups and Kgroups revisited

Section 42.57: Rational intersection products on regular schemes

Section 42.58: Gysin maps for local complete intersection morphisms

Section 42.59: Gysin maps for diagonals

Section 42.60: Exterior product

Section 42.61: Intersection products

Section 42.62: Exterior product over Dedekind domains

Section 42.63: Intersection products over Dedekind domains

Section 42.64: Todd classes

Section 42.65: GrothendieckRiemannRoch

Section 42.66: Change of base scheme

Section 42.67: Appendix A: Alternative approach to key lemma

Section 42.68: Appendix B: Alternative approaches

Chapter 43: Intersection Theory

Section 43.1: Introduction

Section 43.2: Conventions

Section 43.3: Cycles

Section 43.4: Cycle associated to closed subscheme

Section 43.5: Cycle associated to a coherent sheaf

Section 43.6: Proper pushforward

Section 43.7: Flat pullback

Section 43.8: Rational Equivalence

Section 43.9: Rational equivalence and rational functions

Section 43.10: Proper pushforward and rational equivalence

Section 43.11: Flat pullback and rational equivalence

Section 43.12: The short exact sequence for an open

Section 43.13: Proper intersections

Section 43.14: Intersection multiplicities using Tor formula

Section 43.15: Algebraic multiplicities

Section 43.16: Computing intersection multiplicities

Section 43.17: Intersection product using Tor formula

Section 43.18: Exterior product

Section 43.19: Reduction to the diagonal

Section 43.20: Associativity of intersections

Section 43.21: Flat pullback and intersection products

Section 43.22: Projection formula for flat proper morphisms

Section 43.23: Projections

Section 43.24: Moving Lemma

Section 43.25: Intersection products and rational equivalence

Section 43.26: Chow rings

Section 43.27: Pullback for a general morphism

Section 43.28: Pullback of cycles

Chapter 44: Picard Schemes of Curves

Section 44.1: Introduction

Section 44.2: Hilbert scheme of points

Section 44.3: Moduli of divisors on smooth curves

Section 44.4: The Picard functor

Section 44.5: A representability criterion

Section 44.6: The Picard scheme of a curve

Section 44.7: Some remarks on Picard groups

Chapter 45: Weil Cohomology Theories

Section 45.1: Introduction

Section 45.2: Conventions and notation

Section 45.3: Correspondences

Section 45.4: Chow motives

Section 45.5: Chow groups of motives

Section 45.6: Projective space bundle formula

Section 45.7: Classical Weil cohomology theories

Section 45.8: Cycles over nonclosed fields

Section 45.9: Weil cohomology theories, I

Section 45.10: Further properties

Section 45.11: Weil cohomology theories, II

Section 45.12: Chern classes

Section 45.13: Exterior powers and Kgroups

Section 45.14: Weil cohomology theories, III

Chapter 46: Adequate Modules

Section 46.1: Introduction

Section 46.2: Conventions

Section 46.3: Adequate functors

Section 46.4: Higher exts of adequate functors

Section 46.5: Adequate modules

Section 46.6: Parasitic adequate modules

Section 46.7: Derived categories of adequate modules, I

Section 46.8: Pure extensions

Section 46.9: Higher exts of quasicoherent sheaves on the big site

Section 46.10: Derived categories of adequate modules, II

Chapter 47: Dualizing Complexes

Section 47.1: Introduction

Section 47.2: Essential surjections and injections

Section 47.3: Injective modules

Section 47.4: Projective covers

Section 47.5: Injective hulls

Section 47.6: Duality over Artinian local rings

Section 47.7: Injective hull of the residue field

Section 47.8: Deriving torsion

Section 47.9: Local cohomology

Section 47.10: Local cohomology for Noetherian rings

Section 47.11: Depth

Section 47.12: Torsion versus complete modules

Section 47.13: Trivial duality for a ring map

Section 47.14: Base change for trivial duality

Section 47.15: Dualizing complexes

Section 47.16: Dualizing complexes over local rings

Section 47.17: Dualizing complexes and dimension functions

Section 47.18: The local duality theorem

Section 47.19: Dualizing modules

Section 47.20: CohenMacaulay rings

Section 47.21: Gorenstein rings

Section 47.22: The ubiquity of dualizing complexes

Section 47.23: Formal fibres

Section 47.24: Upper shriek algebraically

Section 47.25: Relative dualizing complexes in the Noetherian case

Section 47.26: More on dualizing complexes

Section 47.27: Relative dualizing complexes

Chapter 48: Duality for Schemes

Section 48.1: Introduction

Section 48.2: Dualizing complexes on schemes

Section 48.3: Right adjoint of pushforward

Section 48.4: Right adjoint of pushforward and restriction to opens

Section 48.5: Right adjoint of pushforward and base change, I

Section 48.6: Right adjoint of pushforward and base change, II

Section 48.7: Right adjoint of pushforward and trace maps

Section 48.8: Right adjoint of pushforward and pullback

Section 48.9: Right adjoint of pushforward for closed immersions

Section 48.10: Right adjoint of pushforward for closed immersions and base change

Section 48.11: Right adjoint of pushforward for finite morphisms

Section 48.12: Right adjoint of pushforward for proper flat morphisms

Section 48.13: Right adjoint of pushforward for perfect proper morphisms

Section 48.14: Right adjoint of pushforward for effective Cartier divisors

Section 48.15: Right adjoint of pushforward in examples

Section 48.16: Upper shriek functors

Section 48.17: Properties of upper shriek functors

Section 48.18: Base change for upper shriek

Section 48.19: A duality theory

Section 48.20: Glueing dualizing complexes

Section 48.21: Dimension functions

Section 48.22: Dualizing modules

Section 48.23: CohenMacaulay schemes

Section 48.24: Gorenstein schemes

Section 48.25: Gorenstein morphisms

Section 48.26: More on dualizing complexes

Section 48.27: Duality for proper schemes over fields

Section 48.28: Relative dualizing complexes

Section 48.29: The fundamental class of an lci morphism

Chapter 49: Discriminants and Differents

Section 49.1: Introduction

Section 49.2: Dualizing modules for quasifinite ring maps

Section 49.3: Discriminant of a finite locally free morphism

Section 49.4: Traces for flat quasifinite ring maps

Section 49.5: Finite morphisms

Section 49.6: The Noether different

Section 49.7: The Kähler different

Section 49.8: The Dedekind different

Section 49.9: The different

Section 49.10: Quasifinite syntomic morphisms

Section 49.11: Finite syntomic morphisms

Section 49.12: A formula for the different

Section 49.13: The Tate map

Section 49.14: A generalization of the different

Section 49.15: Comparison with duality theory

Section 49.16: Quasifinite Gorenstein morphisms

Chapter 50: de Rham Cohomology

Section 50.1: Introduction

Section 50.2: The de Rham complex

Section 50.3: de Rham cohomology

Section 50.4: Cup product

Section 50.5: Hodge cohomology

Section 50.6: Two spectral sequences

Section 50.7: The Hodge filtration

Section 50.8: Künneth formula

Section 50.9: First chern class in de Rham cohomology

Section 50.10: de Rham cohomology of a line bundle

Section 50.11: de Rham cohomology of projective space

Section 50.12: The spectral sequence for a smooth morphism

Section 50.13: LerayHirsch type theorems

Section 50.14: Projective space bundle formula

Section 50.15: Log poles along a divisor

Section 50.16: Calculations

Section 50.17: Blowing up and de Rham cohomology

Section 50.18: Comparing sheaves of differential forms

Section 50.19: Trace maps on de Rham complexes

Section 50.20: Poincaré duality

Section 50.21: Chern classes

Section 50.22: A Weil cohomology theory

Chapter 51: Local Cohomology

Section 51.1: Introduction

Section 51.2: Generalities

Section 51.3: Hartshorne's connectedness lemma

Section 51.4: Cohomological dimension

Section 51.5: More general supports

Section 51.6: Filtrations on local cohomology

Section 51.7: Finiteness of local cohomology, I

Section 51.8: Finiteness of pushforwards, I

Section 51.9: Depth and dimension

Section 51.10: Annihilators of local chomology, I

Section 51.11: Finiteness of local cohomology, II

Section 51.12: Finiteness of pushforwards, II

Section 51.13: Annihilators of local chomology, II

Section 51.14: Finiteness of local cohomology, III

Section 51.15: Improving coherent modules

Section 51.16: HartshorneLichtenbaum vanishing

Section 51.17: Frobenius action

Section 51.18: Structure of certain modules

Section 51.19: Additional structure on local cohomology

Chapter 52: Algebraic and Formal Geometry

Section 52.1: Introduction

Section 52.2: Formal sections, I

Section 52.3: Formal sections, II

Section 52.4: Formal sections, III

Section 52.5: MittagLeffler conditions

Section 52.6: Derived completion on a ringed site

Section 52.7: The theorem on formal functions

Section 52.8: Algebraization of local cohomology, I

Section 52.9: Algebraization of local cohomology, II

Section 52.10: Algebraization of local cohomology, III

Section 52.11: Algebraization of formal sections, I

Section 52.12: Algebraization of formal sections, II

Section 52.13: Algebraization of formal sections, III

Section 52.14: Application to connectedness

Section 52.15: The completion functor

Section 52.16: Algebraization of coherent formal modules, I

Section 52.17: Algebraization of coherent formal modules, II

Section 52.18: A distance function

Section 52.19: Algebraization of coherent formal modules, III

Section 52.20: Algebraization of coherent formal modules, IV

Section 52.21: Improving coherent formal modules

Section 52.22: Algebraization of coherent formal modules, V

Section 52.23: Algebraization of coherent formal modules, VI

Section 52.24: Application to the completion functor

Section 52.25: Coherent triples

Section 52.26: Invertible modules on punctured spectra, I

Section 52.27: Invertible modules on punctured spectra, II

Section 52.28: Application to Lefschetz theorems

Chapter 53: Algebraic Curves

Section 53.1: Introduction

Section 53.2: Curves and function fields

Section 53.3: Linear series

Section 53.4: Duality

Section 53.5: RiemannRoch

Section 53.6: Some vanishing results

Section 53.7: Very ample invertible sheaves

Section 53.8: The genus of a curve

Section 53.9: Plane curves

Section 53.10: Curves of genus zero

Section 53.11: Geometric genus

Section 53.12: RiemannHurwitz

Section 53.13: Inseparable maps

Section 53.14: Pushouts

Section 53.15: Glueing and squishing

Section 53.16: Multicross and nodal singularities

Section 53.17: Torsion in the Picard group

Section 53.18: Genus versus geometric genus

Section 53.19: Nodal curves

Section 53.20: Families of nodal curves

Section 53.21: More vanishing results

Section 53.22: Contracting rational tails

Section 53.23: Contracting rational bridges

Section 53.24: Contracting to a stable curve

Section 53.25: Vector fields

Chapter 54: Resolution of Surfaces

Section 54.1: Introduction

Section 54.2: A trace map in positive characteristic

Section 54.3: Quadratic transformations

Section 54.4: Dominating by quadratic transformations

Section 54.5: Dominating by normalized blowups

Section 54.6: Modifying over local rings

Section 54.7: Vanishing

Section 54.8: Boundedness

Section 54.9: Rational singularities

Section 54.10: Formal arcs

Section 54.11: Base change to the completion

Section 54.12: Rational double points

Section 54.13: Implied properties

Section 54.14: Resolution

Section 54.15: Embedded resolution

Section 54.16: Contracting exceptional curves

Section 54.17: Factorization birational maps

Chapter 55: Semistable Reduction

Section 55.1: Introduction

Section 55.2: Linear algebra

Section 55.3: Numerical types

Section 55.4: The Picard group of a numerical type

Section 55.5: Classification of proper subgraphs

Section 55.6: Classification of minimal type for genus zero and one

Section 55.7: Bounding invariants of numerical types

Section 55.8: Models

Section 55.9: The geometry of a regular model

Section 55.10: Uniqueness of the minimal model

Section 55.11: A formula for the genus

Section 55.12: Blowing down exceptional curves

Section 55.13: Picard groups of models

Section 55.14: Semistable reduction

Section 55.15: Semistable reduction in genus zero

Section 55.16: Semistable reduction in genus one

Section 55.17: Semistable reduction in genus at least two

Section 55.18: Semistable reduction for curves

Chapter 56: Fundamental Groups of Schemes

Section 56.1: Introduction

Section 56.2: Schemes étale over a point

Section 56.3: Galois categories

Section 56.4: Functors and homomorphisms

Section 56.5: Finite étale morphisms

Section 56.6: Fundamental groups

Section 56.7: Galois covers of connected schemes

Section 56.8: Topological invariance of the fundamental group

Section 56.9: Finite étale covers of proper schemes

Section 56.10: Local connectedness

Section 56.11: Fundamental groups of normal schemes

Section 56.12: Group actions and integral closure

Section 56.13: Ramification theory

Section 56.14: Geometric and arithmetic fundamental groups

Section 56.15: Homotopy exact sequence

Section 56.16: Specialization maps

Section 56.17: Restriction to a closed subscheme

Section 56.18: Pushouts and fundamental groups

Section 56.19: Finite étale covers of punctured spectra, I

Section 56.20: Purity in local case, I

Section 56.21: Purity of branch locus

Section 56.22: Finite étale covers of punctured spectra, II

Section 56.23: Finite étale covers of punctured spectra, III

Section 56.24: Finite étale covers of punctured spectra, IV

Section 56.25: Purity in local case, II

Section 56.26: Purity in local case, III

Section 56.27: Lefschetz for the fundamental group

Section 56.28: Purity of ramification locus

Section 56.29: Affineness of complement of ramification locus

Section 56.30: Specialization maps in the smooth proper case

Section 56.31: Tame ramification

Chapter 57: Étale Cohomology

Section 57.1: Introduction

Section 57.2: Which sections to skip on a first reading?

Section 57.3: Prologue

Section 57.4: The étale topology

Section 57.5: Feats of the étale topology

Section 57.6: A computation

Section 57.7: Nontorsion coefficients

Section 57.8: Sheaf theory

Section 57.9: Presheaves

Section 57.10: Sites

Section 57.11: Sheaves

Section 57.12: The example of Gsets

Section 57.13: Sheafification

Section 57.14: Cohomology

Section 57.15: The fpqc topology

Section 57.16: Faithfully flat descent

Section 57.17: Quasicoherent sheaves

Section 57.18: Čech cohomology

Section 57.19: The Čechtocohomology spectral sequence

Section 57.20: Big and small sites of schemes

Section 57.21: The étale topos

Section 57.22: Cohomology of quasicoherent sheaves

Section 57.23: Examples of sheaves

Section 57.24: Picard groups

Section 57.25: The étale site

Section 57.26: Étale morphisms

Section 57.27: Étale coverings

Section 57.28: Kummer theory

Section 57.29: Neighborhoods, stalks and points

Section 57.30: Points in other topologies

Section 57.31: Supports of abelian sheaves

Section 57.32: Henselian rings

Section 57.33: Stalks of the structure sheaf

Section 57.34: Functoriality of small étale topos

Section 57.35: Direct images

Section 57.36: Inverse image

Section 57.37: Functoriality of big topoi

Section 57.38: Functoriality and sheaves of modules

Section 57.39: Comparing topologies

Section 57.40: Recovering morphisms

Section 57.41: Push and pull

Section 57.42: Property (A)

Section 57.43: Property (B)

Section 57.44: Property (C)

Section 57.45: Topological invariance of the small étale site

Section 57.46: Closed immersions and pushforward

Section 57.47: Integral universally injective morphisms

Section 57.48: Big sites and pushforward

Section 57.49: Exactness of big lower shriek

Section 57.50: Étale cohomology

Section 57.51: Colimits

Section 57.52: Stalks of higher direct images

Section 57.53: The Leray spectral sequence

Section 57.54: Vanishing of finite higher direct images

Section 57.55: Galois action on stalks

Section 57.56: Group cohomology

Section 57.57: Tate's continuous cohomology

Section 57.58: Cohomology of a point

Section 57.59: Cohomology of curves

Section 57.60: Brauer groups

Section 57.61: The Brauer group of a scheme

Section 57.62: The ArtinSchreier sequence

Section 57.63: Locally constant sheaves

Section 57.64: Locally constant sheaves and the fundamental group

Section 57.65: Méthode de la trace

Section 57.66: Galois cohomology

Section 57.67: Higher vanishing for the multiplicative group

Section 57.68: Picard groups of curves

Section 57.69: Extension by zero

Section 57.70: Constructible sheaves

Section 57.71: Auxiliary lemmas on morphisms

Section 57.72: More on constructible sheaves

Section 57.73: Constructible sheaves on Noetherian schemes

Section 57.74: Complexes with constructible cohomology

Section 57.75: Tor finite with constructible cohomology

Section 57.76: Torsion sheaves

Section 57.77: Cohomology with support in a closed subscheme

Section 57.78: Schemes with strictly henselian local rings

Section 57.79: Affine analog of proper base change

Section 57.80: Cohomology of torsion sheaves on curves

Section 57.81: First cohomology of proper schemes

Section 57.82: Preliminaries on base change

Section 57.83: Base change for pushforward

Section 57.84: Base change for higher direct images

Section 57.85: Smooth base change

Section 57.86: Applications of smooth base change

Section 57.87: The proper base change theorem

Section 57.88: Applications of proper base change

Section 57.89: Cohomological dimension

Section 57.90: Finite cohomological dimension

Section 57.91: Künneth in étale cohomology

Section 57.92: Comparing chaotic and Zariski topologies

Section 57.93: Comparing big and small topoi

Section 57.94: Comparing fppf and étale topologies

Section 57.95: Comparing fppf and étale topologies: modules

Section 57.96: Comparing ph and étale topologies

Section 57.97: Comparing h and étale topologies

Section 57.98: Blow up squares and étale cohomology

Section 57.99: Almost blow up squares and the h topology

Section 57.100: Cohomology of the structure sheaf in the h topology

Chapter 58: Crystalline Cohomology

Section 58.1: Introduction

Section 58.2: Divided power envelope

Section 58.3: Some explicit divided power thickenings

Section 58.4: Compatibility

Section 58.5: Affine crystalline site

Section 58.6: Module of differentials

Section 58.7: Divided power schemes

Section 58.8: The big crystalline site

Section 58.9: The crystalline site

Section 58.10: Sheaves on the crystalline site

Section 58.11: Crystals in modules

Section 58.12: Sheaf of differentials

Section 58.13: Two universal thickenings

Section 58.14: The de Rham complex

Section 58.15: Connections

Section 58.16: Cosimplicial algebra

Section 58.17: Crystals in quasicoherent modules

Section 58.18: General remarks on cohomology

Section 58.19: Cosimplicial preparations

Section 58.20: Divided power Poincaré lemma

Section 58.21: Cohomology in the affine case

Section 58.22: Two counter examples

Section 58.23: Applications

Section 58.24: Some further results

Section 58.25: Pulling back along purely inseparable maps

Section 58.26: Frobenius action on crystalline cohomology

Chapter 59: Proétale Cohomology

Section 59.1: Introduction

Section 59.2: Some topology

Section 59.3: Local isomorphisms

Section 59.4: IndZariski algebra

Section 59.5: Constructing wlocal affine schemes

Section 59.6: Identifying local rings versus indZariski

Section 59.7: Indétale algebra

Section 59.8: Constructing indétale algebras

Section 59.9: Weakly étale versus proétale

Section 59.10: The V topology and the proh topology

Section 59.11: Constructing wcontractible covers

Section 59.12: The proétale site

Section 59.13: Weakly contractible objects

Section 59.14: Weakly contractible hypercoverings

Section 59.15: Compact generation

Section 59.16: Comparing topologies

Section 59.17: Comparing big and small topoi

Section 59.18: Points of the proétale site

Section 59.19: Comparison with the étale site

Section 59.20: Derived completion in the constant Noetherian case

Section 59.21: Derived completion and weakly contractible objects

Section 59.22: Cohomology of a point

Section 59.23: Functoriality of the proétale site

Section 59.24: Finite morphisms and proétale sites

Section 59.25: Closed immersions and proétale sites

Section 59.26: Extension by zero

Section 59.27: Constructible sheaves on the proétale site

Section 59.28: Constructible adic sheaves

Section 59.29: A suitable derived category

Section 59.30: Proper base change

Section 59.31: Change of partial universe

Chapter 60: More Étale Cohomology

Section 60.1: Introduction

Section 60.2: Growing sections

Section 60.3: Sections with compact support

Section 60.4: Sections with finite support

Section 60.5: Upper shriek for locally quasifinite morphisms

Section 60.6: Derived upper shriek for locally quasifinite morphisms

Section 60.7: Preliminaries to derived lower shriek via compactifications

Section 60.8: Derived lower shriek via compactifications

Chapter 61: The Trace Formula

Section 61.1: Introduction

Section 61.2: The trace formula

Section 61.3: Frobenii

Section 61.4: Traces

Section 61.5: Why derived categories?

Section 61.6: Derived categories

Section 61.7: Filtered derived category

Section 61.8: Filtered derived functors

Section 61.9: Application of filtered complexes

Section 61.10: Perfectness

Section 61.11: Filtrations and perfect complexes

Section 61.12: Characterizing perfect objects

Section 61.13: Cohomology of nice complexes

Section 61.14: Lefschetz numbers

Section 61.15: Preliminaries and sorites

Section 61.16: Proof of the trace formula

Section 61.17: Applications

Section 61.18: On ladic sheaves

Section 61.19: Lfunctions

Section 61.20: Cohomological interpretation

Section 61.21: List of things which we should add above

Section 61.22: Examples of Lfunctions

Section 61.23: Constant sheaves

Section 61.24: The Legendre family

Section 61.25: Exponential sums

Section 61.26: Trace formula in terms of fundamental groups

Section 61.27: Fundamental groups

Section 61.28: Profinite groups, cohomology and homology

Section 61.29: Cohomology of curves, revisited

Section 61.30: Abstract trace formula

Section 61.31: Automorphic forms and sheaves

Section 61.32: Counting points

Section 61.33: Precise form of Chebotarev

Section 61.34: How many primes decompose completely?

Section 61.35: How many points are there really?