The Stacks project

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 envelopes
    • Section 42.23: Chow groups and K-groups
    • Section 42.24: The divisor associated to an invertible sheaf
    • Section 42.25: Intersecting with an invertible sheaf
    • Section 42.26: Intersecting with an invertible sheaf and push and pull
    • Section 42.27: The key formula
    • Section 42.28: Intersecting with an invertible sheaf and rational equivalence
    • Section 42.29: Gysin homomorphisms
    • Section 42.30: Gysin homomorphisms and rational equivalence
    • Section 42.31: Relative effective Cartier divisors
    • Section 42.32: Affine bundles
    • Section 42.33: Bivariant intersection theory
    • Section 42.34: Chow cohomology and the first Chern class
    • Section 42.35: Lemmas on bivariant classes
    • Section 42.36: Projective space bundle formula
    • Section 42.37: The Chern classes of a vector bundle
    • Section 42.38: Intersecting with Chern classes
    • Section 42.39: Polynomial relations among Chern classes
    • Section 42.40: Additivity of Chern classes
    • Section 42.41: Degrees of zero cycles
    • Section 42.42: Cycles of given codimension
    • Section 42.43: The splitting principle
    • Section 42.44: Chern classes and sections
    • Section 42.45: The Chern character and tensor products
    • Section 42.46: Chern classes and the derived category
    • Section 42.47: A baby case of localized Chern classes
    • Section 42.48: Gysin at infinity
    • Section 42.49: Preparation for localized Chern classes
    • Section 42.50: Localized Chern classes
    • Section 42.51: Two technical lemmas
    • Section 42.52: Properties of localized Chern classes
    • Section 42.53: Blowing up at infinity
    • Section 42.54: Higher codimension gysin homomorphisms
    • Section 42.55: Calculating some classes
    • Section 42.56: An Adams operator
    • Section 42.57: Chow groups and K-groups revisited
    • Section 42.58: Rational intersection products on regular schemes
    • Section 42.59: Gysin maps for local complete intersection morphisms
    • Section 42.60: Gysin maps for diagonals
    • Section 42.61: Exterior product
    • Section 42.62: Intersection products
    • Section 42.63: Exterior product over Dedekind domains
    • Section 42.64: Intersection products over Dedekind domains
    • Section 42.65: Todd classes
    • Section 42.66: Grothendieck-Riemann-Roch
    • Section 42.67: Change of base scheme
    • Section 42.68: Appendix A: Alternative approach to key lemma
    • Section 42.69: 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
  • 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 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 54.3: 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: Functors and Morphisms
    • Section 56.1: Introduction
    • Section 56.2: Functors on module categories
    • Section 56.3: Functors between categories of modules
    • Section 56.4: Extending functors on categories of modules
    • Section 56.5: Functors between categories of quasi-coherent modules
    • Section 56.6: Gabriel-Rosenberg reconstruction
    • Section 56.7: Functors between categories of coherent modules
  • Chapter 57: Derived Categories of Varieties
    • Section 57.1: Introduction
    • Section 57.2: Conventions and notation
    • Section 57.3: Serre functors
    • Section 57.4: Examples of Serre functors
    • Section 57.5: Characterizing coherent modules
    • Section 57.6: A representability theorem
    • Section 57.7: Existence of adjoints
    • Section 57.8: Fourier-Mukai functors
    • Section 57.9: Resolutions and bounds
    • Section 57.10: Sibling functors
    • Section 57.11: Deducing fully faithfulness
    • Section 57.12: Special functors
    • Section 57.13: Fully faithful functors
    • Section 57.14: A category of Fourier-Mukai kernels
    • Section 57.15: Relative equivalences
    • Section 57.16: No deformations
    • Section 57.17: Countability
    • Section 57.18: Countability of derived equivalent varieties
  • Chapter 58: Fundamental Groups of Schemes
    • Section 58.1: Introduction
    • Section 58.2: Schemes étale over a point
    • Section 58.3: Galois categories
    • Section 58.4: Functors and homomorphisms
    • Section 58.5: Finite étale morphisms
    • Section 58.6: Fundamental groups
    • Section 58.7: Galois covers of connected schemes
    • Section 58.8: Topological invariance of the fundamental group
    • Section 58.9: Finite étale covers of proper schemes
    • Section 58.10: Local connectedness
    • Section 58.11: Fundamental groups of normal schemes
    • Section 58.12: Group actions and integral closure
    • Section 58.13: Ramification theory
    • Section 58.14: Geometric and arithmetic fundamental groups
    • Section 58.15: Homotopy exact sequence
    • Section 58.16: Specialization maps
    • Section 58.17: Restriction to a closed subscheme
    • Section 58.18: Pushouts and fundamental groups
    • Section 58.19: Finite étale covers of punctured spectra, I
    • Section 58.20: Purity in local case, I
    • Section 58.21: Purity of branch locus
    • Section 58.22: Finite étale covers of punctured spectra, II
    • Section 58.23: Finite étale covers of punctured spectra, III
    • Section 58.24: Finite étale covers of punctured spectra, IV
    • Section 58.25: Purity in local case, II
    • Section 58.26: Purity in local case, III
    • Section 58.27: Lefschetz for the fundamental group
    • Section 58.28: Purity of ramification locus
    • Section 58.29: Affineness of complement of ramification locus
    • Section 58.30: Specialization maps in the smooth proper case
    • Section 58.31: Tame ramification
    • Section 58.32: Tricks in positive characteristic
  • Chapter 59: Étale Cohomology
    • Section 59.1: Introduction
    • Section 59.2: Which sections to skip on a first reading?
    • Section 59.3: Prologue
    • Section 59.4: The étale topology
    • Section 59.5: Feats of the étale topology
    • Section 59.6: A computation
    • Section 59.7: Nontorsion coefficients
    • Section 59.8: Sheaf theory
    • Section 59.9: Presheaves
    • Section 59.10: Sites
    • Section 59.11: Sheaves
    • Section 59.12: The example of G-sets
    • Section 59.13: Sheafification
    • Section 59.14: Cohomology
    • Section 59.15: The fpqc topology
    • Section 59.16: Faithfully flat descent
    • Section 59.17: Quasi-coherent sheaves
    • Section 59.18: Čech cohomology
    • Section 59.19: The Čech-to-cohomology spectral sequence
    • Section 59.20: Big and small sites of schemes
    • Section 59.21: The étale topos
    • Section 59.22: Cohomology of quasi-coherent sheaves
    • Section 59.23: Examples of sheaves
    • Section 59.24: Picard groups
    • Section 59.25: The étale site
    • Section 59.26: Étale morphisms
    • Section 59.27: Étale coverings
    • Section 59.28: Kummer theory
    • Section 59.29: Neighborhoods, stalks and points
    • Section 59.30: Points in other topologies
    • Section 59.31: Supports of abelian sheaves
    • Section 59.32: Henselian rings
    • Section 59.33: Stalks of the structure sheaf
    • Section 59.34: Functoriality of small étale topos
    • Section 59.35: Direct images
    • Section 59.36: Inverse image
    • Section 59.37: Functoriality of big topoi
    • Section 59.38: Functoriality and sheaves of modules
    • Section 59.39: Comparing topologies
    • Section 59.40: Recovering morphisms
    • Section 59.41: Push and pull
    • Section 59.42: Property (A)
    • Section 59.43: Property (B)
    • Section 59.44: Property (C)
    • Section 59.45: Topological invariance of the small étale site
    • Section 59.46: Closed immersions and pushforward
    • Section 59.47: Integral universally injective morphisms
    • Section 59.48: Big sites and pushforward
    • Section 59.49: Exactness of big lower shriek
    • Section 59.50: Étale cohomology
    • Section 59.51: Colimits
    • Section 59.52: Colimits and complexes
    • Section 59.53: Stalks of higher direct images
    • Section 59.54: The Leray spectral sequence
    • Section 59.55: Vanishing of finite higher direct images
    • Section 59.56: Galois action on stalks
    • Section 59.57: Group cohomology
    • Section 59.58: Tate's continuous cohomology
    • Section 59.59: Cohomology of a point
    • Section 59.60: Cohomology of curves
    • Section 59.61: Brauer groups
    • Section 59.62: The Brauer group of a scheme
    • Section 59.63: The Artin-Schreier sequence
    • Section 59.64: Locally constant sheaves
    • Section 59.65: Locally constant sheaves and the fundamental group
    • Section 59.66: Méthode de la trace
    • Section 59.67: Galois cohomology
    • Section 59.68: Higher vanishing for the multiplicative group
    • Section 59.69: Picard groups of curves
    • Section 59.70: Extension by zero
    • Section 59.71: Constructible sheaves
    • Section 59.72: Auxiliary lemmas on morphisms
    • Section 59.73: More on constructible sheaves
    • Section 59.74: Constructible sheaves on Noetherian schemes
    • Section 59.75: Specializations and étale sheaves
    • Section 59.76: Complexes with constructible cohomology
    • Section 59.77: Tor finite with constructible cohomology
    • Section 59.78: Torsion sheaves
    • Section 59.79: Cohomology with support in a closed subscheme
    • Section 59.80: Schemes with strictly henselian local rings
    • Section 59.81: Absolutely integrally closed vanishing
    • Section 59.82: Affine analog of proper base change
    • Section 59.83: Cohomology of torsion sheaves on curves
    • Section 59.84: Cohomology of torsion modules on curves
    • Section 59.85: First cohomology of proper schemes
    • Section 59.86: Preliminaries on base change
    • Section 59.87: Base change for pushforward
    • Section 59.88: Base change for higher direct images
    • Section 59.89: Smooth base change
    • Section 59.90: Applications of smooth base change
    • Section 59.91: The proper base change theorem
    • Section 59.92: Applications of proper base change
    • Section 59.93: Local acyclicity
    • Section 59.94: The cospecialization map
    • Section 59.95: Cohomological dimension
    • Section 59.96: Finite cohomological dimension
    • Section 59.97: Künneth in étale cohomology
    • Section 59.98: Comparing chaotic and Zariski topologies
    • Section 59.99: Comparing big and small topoi
    • Section 59.100: Comparing fppf and étale topologies
    • Section 59.101: Comparing fppf and étale topologies: modules
    • Section 59.102: Comparing ph and étale topologies
    • Section 59.103: Comparing h and étale topologies
    • Section 59.104: Descending étale sheaves
    • Section 59.105: Blow up squares and étale cohomology
    • Section 59.106: Almost blow up squares and the h topology
    • Section 59.107: Cohomology of the structure sheaf in the h topology
  • Chapter 60: Crystalline Cohomology
    • Section 60.1: Introduction
    • Section 60.2: Divided power envelope
    • Section 60.3: Some explicit divided power thickenings
    • Section 60.4: Compatibility
    • Section 60.5: Affine crystalline site
    • Section 60.6: Module of differentials
    • Section 60.7: Divided power schemes
    • Section 60.8: The big crystalline site
    • Section 60.9: The crystalline site
    • Section 60.10: Sheaves on the crystalline site
    • Section 60.11: Crystals in modules
    • Section 60.12: Sheaf of differentials
    • Section 60.13: Two universal thickenings
    • Section 60.14: The de Rham complex
    • Section 60.15: Connections
    • Section 60.16: Cosimplicial algebra
    • Section 60.17: Crystals in quasi-coherent modules
    • Section 60.18: General remarks on cohomology
    • Section 60.19: Cosimplicial preparations
    • Section 60.20: Divided power Poincaré lemma
    • Section 60.21: Cohomology in the affine case
    • Section 60.22: Two counter examples
    • Section 60.23: Applications
    • Section 60.24: Some further results
    • Section 60.25: Pulling back along purely inseparable maps
    • Section 60.26: Frobenius action on crystalline cohomology
  • Chapter 61: Pro-étale Cohomology
    • Section 61.1: Introduction
    • Section 61.2: Some topology
    • Section 61.3: Local isomorphisms
    • Section 61.4: Ind-Zariski algebra
    • Section 61.5: Constructing w-local affine schemes
    • Section 61.6: Identifying local rings versus ind-Zariski
    • Section 61.7: Ind-étale algebra
    • Section 61.8: Constructing ind-étale algebras
    • Section 61.9: Weakly étale versus pro-étale
    • Section 61.10: The V topology and the pro-h topology
    • Section 61.11: Constructing w-contractible covers
    • Section 61.12: The pro-étale site
    • Section 61.13: Weakly contractible objects
    • Section 61.14: Weakly contractible hypercoverings
    • Section 61.15: Compact generation
    • Section 61.16: Comparing topologies
    • Section 61.17: Comparing big and small topoi
    • Section 61.18: Points of the pro-étale site
    • Section 61.19: Comparison with the étale site
    • Section 61.20: Derived completion in the constant Noetherian case
    • Section 61.21: Derived completion and weakly contractible objects
    • Section 61.22: Cohomology of a point
    • Section 61.23: Functoriality of the pro-étale site
    • Section 61.24: Finite morphisms and pro-étale sites
    • Section 61.25: Closed immersions and pro-étale sites
    • Section 61.26: Extension by zero
    • Section 61.27: Constructible sheaves on the pro-étale site
    • Section 61.28: Constructible adic sheaves
    • Section 61.29: A suitable derived category
    • Section 61.30: Proper base change
    • Section 61.31: Change of partial universe
  • Chapter 62: Relative Cycles
    • Section 62.1: Introduction
    • Section 62.2: Conventions and notation
    • Section 62.3: Cycles relative to fields
    • Section 62.4: Specialization of cycles
    • Section 62.5: Families of cycles on fibres
    • Section 62.6: Relative cycles
    • Section 62.7: Equidimensional relative cycles
    • Section 62.8: Effective relative cycles
    • Section 62.9: Proper relative cycles
    • Section 62.10: Proper and equidimensional relative cycles
    • Section 62.11: Action on cycles
    • Section 62.12: Action on chow groups
    • Section 62.13: Composition of families of cycles on fibres
    • Section 62.14: Composition of relative cycles
    • Section 62.15: Comparison with Suslin and Voevodsky
    • Section 62.16: Relative cycles in the non-Noetherian case
  • Chapter 63: More Étale Cohomology
    • Section 63.1: Introduction
    • Section 63.2: Growing sections
    • Section 63.3: Sections with compact support
    • Section 63.4: Sections with finite support
    • Section 63.5: Weightings and trace maps for locally quasi-finite morphisms
    • Section 63.6: Upper shriek for locally quasi-finite morphisms
    • Section 63.7: Derived upper shriek for locally quasi-finite morphisms
    • Section 63.8: Preliminaries to derived lower shriek via compactifications
    • Section 63.9: Derived lower shriek via compactifications
    • Section 63.10: Properties of derived lower shriek
    • Section 63.11: Derived upper shriek
    • Section 63.12: Compactly supported cohomology
    • Section 63.13: A constructibility result
    • Section 63.14: Complexes with constructible cohomology
    • Section 63.15: Applications
    • Section 63.16: More on derived upper shriek
  • Chapter 64: The Trace Formula
    • Section 64.1: Introduction
    • Section 64.2: The trace formula
    • Section 64.3: Frobenii
    • Section 64.4: Traces
    • Section 64.5: Why derived categories?
    • Section 64.6: Derived categories
    • Section 64.7: Filtered derived category
    • Section 64.8: Filtered derived functors
    • Section 64.9: Application of filtered complexes
    • Section 64.10: Perfectness
    • Section 64.11: Filtrations and perfect complexes
    • Section 64.12: Characterizing perfect objects
    • Section 64.13: Cohomology of nice complexes
    • Section 64.14: Lefschetz numbers
    • Section 64.15: Preliminaries and sorites
    • Section 64.16: Proof of the trace formula
    • Section 64.17: Applications
    • Section 64.18: On l-adic sheaves
    • Section 64.19: L-functions
    • Section 64.20: Cohomological interpretation
    • Section 64.21: List of things which we should add above
    • Section 64.22: Examples of L-functions
    • Section 64.23: Constant sheaves
    • Section 64.24: The Legendre family
    • Section 64.25: Exponential sums
    • Section 64.26: Trace formula in terms of fundamental groups
    • Section 64.27: Fundamental groups
    • Section 64.28: Profinite groups, cohomology and homology
    • Section 64.29: Cohomology of curves, revisited
    • Section 64.30: Abstract trace formula
    • Section 64.31: Automorphic forms and sheaves
    • Section 64.32: Counting points
    • Section 64.33: Precise form of Chebotarev
    • Section 64.34: How many primes decompose completely?
    • Section 64.35: How many points are there really?