# 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 K-groups
• 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 K-groups 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: Grothendieck-Riemann-Roch
• 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 non-closed 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 K-groups
• Section 45.14: Weil cohomology theories, III
• Section 46.1: Introduction
• Section 46.2: Conventions
• Section 46.4: Higher exts of adequate functors
• 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 quasi-coherent 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: Cohen-Macaulay 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: Cohen-Macaulay 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
• Section 48.30: Extension by zero for coherent modules
• Section 48.31: Preliminaries to compactly supported cohomology
• Section 48.32: Compactly supported cohomology for coherent modules
• Section 48.33: Duality for compactly supported cohomology
• Section 48.34: Lichtenbaum's theorem
• Chapter 49: Discriminants and Differents
• Section 49.1: Introduction
• Section 49.2: Dualizing modules for quasi-finite ring maps
• Section 49.3: Discriminant of a finite locally free morphism
• Section 49.4: Traces for flat quasi-finite 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: Quasi-finite 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: Quasi-finite 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: Leray-Hirsch 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
• Section 50.23: Gysin maps for closed immersions
• Section 50.24: Relative Poincaré duality
• 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 cohomology, I
• Section 51.11: Finiteness of local cohomology, II
• Section 51.12: Finiteness of pushforwards, II
• Section 51.13: Annihilators of local cohomology, II
• Section 51.14: Finiteness of local cohomology, III
• Section 51.15: Improving coherent modules
• Section 51.16: Hartshorne-Lichtenbaum vanishing
• Section 51.17: Frobenius action
• Section 51.18: Structure of certain modules
• Section 51.19: Additional structure on local cohomology
• Section 51.20: A bit of uniformity, I
• Section 51.21: A bit of uniformity, II
• Section 51.22: A bit of uniformity, III
• 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: Mittag-Leffler 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: Riemann-Roch
• 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: Riemann-Hurwitz
• 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 54.4: 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: Derived Categories of Varieties
• Section 56.1: Introduction
• Section 56.2: Conventions and notation
• Section 56.3: Serre functors
• Section 56.4: Examples of Serre functors
• Section 56.5: Characterizing coherent modules
• Section 56.6: A representability theorem
• Section 56.7: Representability in the regular proper case
• Section 56.8: Existence of adjoints
• Section 56.9: Fourier-Mukai functors
• Section 56.10: Resolutions and bounds
• Section 56.11: Functors between categories of quasi-coherent modules
• Section 56.12: Functors between categories of coherent modules
• Section 56.13: Sibling functors
• Section 56.14: Deducing fully faithfulness
• Section 56.15: Special functors
• Section 56.16: Fully faithful functors
• Section 56.17: A category of Fourier-Mukai kernels
• Section 56.18: Relative equivalences
• Section 56.19: No deformations
• Section 56.20: Countability
• Section 56.21: Countability of derived equivalent varieties
• Chapter 57: Fundamental Groups of Schemes
• Section 57.1: Introduction
• Section 57.2: Schemes étale over a point
• Section 57.3: Galois categories
• Section 57.4: Functors and homomorphisms
• Section 57.5: Finite étale morphisms
• Section 57.6: Fundamental groups
• Section 57.7: Galois covers of connected schemes
• Section 57.8: Topological invariance of the fundamental group
• Section 57.9: Finite étale covers of proper schemes
• Section 57.10: Local connectedness
• Section 57.11: Fundamental groups of normal schemes
• Section 57.12: Group actions and integral closure
• Section 57.13: Ramification theory
• Section 57.14: Geometric and arithmetic fundamental groups
• Section 57.15: Homotopy exact sequence
• Section 57.16: Specialization maps
• Section 57.17: Restriction to a closed subscheme
• Section 57.18: Pushouts and fundamental groups
• Section 57.19: Finite étale covers of punctured spectra, I
• Section 57.20: Purity in local case, I
• Section 57.21: Purity of branch locus
• Section 57.22: Finite étale covers of punctured spectra, II
• Section 57.23: Finite étale covers of punctured spectra, III
• Section 57.24: Finite étale covers of punctured spectra, IV
• Section 57.25: Purity in local case, II
• Section 57.26: Purity in local case, III
• Section 57.27: Lefschetz for the fundamental group
• Section 57.28: Purity of ramification locus
• Section 57.29: Affineness of complement of ramification locus
• Section 57.30: Specialization maps in the smooth proper case
• Section 57.31: Tame ramification
• Section 57.32: Tricks in positive characteristic
• Chapter 58: Étale Cohomology
• Section 58.1: Introduction
• Section 58.2: Which sections to skip on a first reading?
• Section 58.3: Prologue
• Section 58.4: The étale topology
• Section 58.5: Feats of the étale topology
• Section 58.6: A computation
• Section 58.7: Nontorsion coefficients
• Section 58.8: Sheaf theory
• Section 58.9: Presheaves
• Section 58.10: Sites
• Section 58.11: Sheaves
• Section 58.12: The example of G-sets
• Section 58.13: Sheafification
• Section 58.14: Cohomology
• Section 58.15: The fpqc topology
• Section 58.16: Faithfully flat descent
• Section 58.17: Quasi-coherent sheaves
• Section 58.18: Čech cohomology
• Section 58.19: The Čech-to-cohomology spectral sequence
• Section 58.20: Big and small sites of schemes
• Section 58.21: The étale topos
• Section 58.22: Cohomology of quasi-coherent sheaves
• Section 58.23: Examples of sheaves
• Section 58.24: Picard groups
• Section 58.25: The étale site
• Section 58.26: Étale morphisms
• Section 58.27: Étale coverings
• Section 58.28: Kummer theory
• Section 58.29: Neighborhoods, stalks and points
• Section 58.30: Points in other topologies
• Section 58.31: Supports of abelian sheaves
• Section 58.32: Henselian rings
• Section 58.33: Stalks of the structure sheaf
• Section 58.34: Functoriality of small étale topos
• Section 58.35: Direct images
• Section 58.36: Inverse image
• Section 58.37: Functoriality of big topoi
• Section 58.38: Functoriality and sheaves of modules
• Section 58.39: Comparing topologies
• Section 58.40: Recovering morphisms
• Section 58.41: Push and pull
• Section 58.42: Property (A)
• Section 58.43: Property (B)
• Section 58.44: Property (C)
• Section 58.45: Topological invariance of the small étale site
• Section 58.46: Closed immersions and pushforward
• Section 58.47: Integral universally injective morphisms
• Section 58.48: Big sites and pushforward
• Section 58.49: Exactness of big lower shriek
• Section 58.50: Étale cohomology
• Section 58.51: Colimits
• Section 58.52: Colimits and complexes
• Section 58.53: Stalks of higher direct images
• Section 58.54: The Leray spectral sequence
• Section 58.55: Vanishing of finite higher direct images
• Section 58.56: Galois action on stalks
• Section 58.57: Group cohomology
• Section 58.58: Tate's continuous cohomology
• Section 58.59: Cohomology of a point
• Section 58.60: Cohomology of curves
• Section 58.61: Brauer groups
• Section 58.62: The Brauer group of a scheme
• Section 58.63: The Artin-Schreier sequence
• Section 58.64: Locally constant sheaves
• Section 58.65: Locally constant sheaves and the fundamental group
• Section 58.66: Méthode de la trace
• Section 58.67: Galois cohomology
• Section 58.68: Higher vanishing for the multiplicative group
• Section 58.69: Picard groups of curves
• Section 58.70: Extension by zero
• Section 58.71: Constructible sheaves
• Section 58.72: Auxiliary lemmas on morphisms
• Section 58.73: More on constructible sheaves
• Section 58.74: Constructible sheaves on Noetherian schemes
• Section 58.75: Specializations and étale sheaves
• Section 58.76: Complexes with constructible cohomology
• Section 58.77: Tor finite with constructible cohomology
• Section 58.78: Torsion sheaves
• Section 58.79: Cohomology with support in a closed subscheme
• Section 58.80: Schemes with strictly henselian local rings
• Section 58.81: Affine analog of proper base change
• Section 58.82: Cohomology of torsion sheaves on curves
• Section 58.83: Cohomology of torsion modules on curves
• Section 58.84: First cohomology of proper schemes
• Section 58.85: Preliminaries on base change
• Section 58.86: Base change for pushforward
• Section 58.87: Base change for higher direct images
• Section 58.88: Smooth base change
• Section 58.89: Applications of smooth base change
• Section 58.90: The proper base change theorem
• Section 58.91: Applications of proper base change
• Section 58.92: Local acyclicity
• Section 58.93: The cospecialization map
• Section 58.94: Cohomological dimension
• Section 58.95: Finite cohomological dimension
• Section 58.96: Künneth in étale cohomology
• Section 58.97: Comparing chaotic and Zariski topologies
• Section 58.98: Comparing big and small topoi
• Section 58.99: Comparing fppf and étale topologies
• Section 58.100: Comparing fppf and étale topologies: modules
• Section 58.101: Comparing ph and étale topologies
• Section 58.102: Comparing h and étale topologies
• Section 58.103: Descending étale sheaves
• Section 58.104: Blow up squares and étale cohomology
• Section 58.105: Almost blow up squares and the h topology
• Section 58.106: Cohomology of the structure sheaf in the h topology
• Chapter 59: Crystalline Cohomology
• Section 59.1: Introduction
• Section 59.2: Divided power envelope
• Section 59.3: Some explicit divided power thickenings
• Section 59.4: Compatibility
• Section 59.5: Affine crystalline site
• Section 59.6: Module of differentials
• Section 59.7: Divided power schemes
• Section 59.8: The big crystalline site
• Section 59.9: The crystalline site
• Section 59.10: Sheaves on the crystalline site
• Section 59.11: Crystals in modules
• Section 59.12: Sheaf of differentials
• Section 59.13: Two universal thickenings
• Section 59.14: The de Rham complex
• Section 59.15: Connections
• Section 59.16: Cosimplicial algebra
• Section 59.17: Crystals in quasi-coherent modules
• Section 59.18: General remarks on cohomology
• Section 59.19: Cosimplicial preparations
• Section 59.20: Divided power Poincaré lemma
• Section 59.21: Cohomology in the affine case
• Section 59.22: Two counter examples
• Section 59.23: Applications
• Section 59.24: Some further results
• Section 59.25: Pulling back along purely inseparable maps
• Section 59.26: Frobenius action on crystalline cohomology
• Chapter 60: Pro-étale Cohomology
• Section 60.1: Introduction
• Section 60.2: Some topology
• Section 60.3: Local isomorphisms
• Section 60.4: Ind-Zariski algebra
• Section 60.5: Constructing w-local affine schemes
• Section 60.6: Identifying local rings versus ind-Zariski
• Section 60.7: Ind-étale algebra
• Section 60.8: Constructing ind-étale algebras
• Section 60.9: Weakly étale versus pro-étale
• Section 60.10: The V topology and the pro-h topology
• Section 60.11: Constructing w-contractible covers
• Section 60.12: The pro-étale site
• Section 60.13: Weakly contractible objects
• Section 60.14: Weakly contractible hypercoverings
• Section 60.15: Compact generation
• Section 60.16: Comparing topologies
• Section 60.17: Comparing big and small topoi
• Section 60.18: Points of the pro-étale site
• Section 60.19: Comparison with the étale site
• Section 60.20: Derived completion in the constant Noetherian case
• Section 60.21: Derived completion and weakly contractible objects
• Section 60.22: Cohomology of a point
• Section 60.23: Functoriality of the pro-étale site
• Section 60.24: Finite morphisms and pro-étale sites
• Section 60.25: Closed immersions and pro-étale sites
• Section 60.26: Extension by zero
• Section 60.27: Constructible sheaves on the pro-étale site
• Section 60.28: Constructible adic sheaves
• Section 60.29: A suitable derived category
• Section 60.30: Proper base change
• Section 60.31: Change of partial universe
• Chapter 61: More Étale Cohomology
• Section 61.1: Introduction
• Section 61.2: Growing sections
• Section 61.3: Sections with compact support
• Section 61.4: Sections with finite support
• Section 61.5: Weightings and trace maps for locally quasi-finite morphisms
• Section 61.6: Upper shriek for locally quasi-finite morphisms
• Section 61.7: Derived upper shriek for locally quasi-finite morphisms
• Section 61.8: Preliminaries to derived lower shriek via compactifications
• Section 61.9: Derived lower shriek via compactifications
• Section 61.10: Properties of derived lower shriek
• Section 61.11: Derived upper shriek
• Section 61.12: Compactly supported cohomology
• Section 61.13: A constructibility result
• Section 61.14: Complexes with constructible cohomology
• Section 61.15: Applications
• Section 61.16: More on derived upper shriek
• Chapter 62: The Trace Formula
• Section 62.1: Introduction
• Section 62.2: The trace formula
• Section 62.3: Frobenii
• Section 62.4: Traces
• Section 62.5: Why derived categories?
• Section 62.6: Derived categories
• Section 62.7: Filtered derived category
• Section 62.8: Filtered derived functors
• Section 62.9: Application of filtered complexes
• Section 62.10: Perfectness
• Section 62.11: Filtrations and perfect complexes
• Section 62.12: Characterizing perfect objects
• Section 62.13: Cohomology of nice complexes
• Section 62.14: Lefschetz numbers
• Section 62.15: Preliminaries and sorites
• Section 62.16: Proof of the trace formula
• Section 62.17: Applications
• Section 62.18: On l-adic sheaves
• Section 62.19: L-functions
• Section 62.20: Cohomological interpretation
• Section 62.21: List of things which we should add above
• Section 62.22: Examples of L-functions
• Section 62.23: Constant sheaves
• Section 62.24: The Legendre family
• Section 62.25: Exponential sums
• Section 62.26: Trace formula in terms of fundamental groups
• Section 62.27: Fundamental groups
• Section 62.28: Profinite groups, cohomology and homology
• Section 62.29: Cohomology of curves, revisited
• Section 62.30: Abstract trace formula
• Section 62.31: Automorphic forms and sheaves
• Section 62.32: Counting points
• Section 62.33: Precise form of Chebotarev
• Section 62.34: How many primes decompose completely?
• Section 62.35: How many points are there really?