The Stacks project

\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} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \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: Grothendieck-Riemann-Roch
    • 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 quasi-coherent 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: Cohen-Macaulay 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: Cohen-Macaulay 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 quasi-finite ring maps
    • Section 47.3: Discriminant of a finite locally free morphism
    • Section 47.4: Traces for flat quasi-finite 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: Quasi-finite 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: Quasi-finite 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: Hartshorne-Lichtenbaum 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: Mittag-Leffler 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: Riemann-Roch
    • 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: Riemann-Hurwitz
    • 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: Topological invariance of the fundamental group
    • Section 53.8: Finite étale covers of proper schemes
    • Section 53.9: Local connectedness
    • Section 53.10: Fundamental groups of normal schemes
    • Section 53.11: Group actions and integral closure
    • Section 53.12: Ramification theory
    • Section 53.13: Geometric and arithmetic fundamental groups
    • Section 53.14: Homotopy exact sequence
    • Section 53.15: Specialization maps
    • Section 53.16: Restriction to a closed subscheme
    • Section 53.17: Pushouts and fundamental groups
    • Section 53.18: Finite étale covers of punctured spectra, I
    • Section 53.19: Purity in local case, I
    • Section 53.20: Purity of branch locus
    • Section 53.21: Finite étale covers of punctured spectra, II
    • Section 53.22: Finite étale covers of punctured spectra, III
    • Section 53.23: Finite étale covers of punctured spectra, IV
    • Section 53.24: Purity in local case, II
    • Section 53.25: Lefschetz for the fundamental group
    • Section 53.26: Purity of ramification locus
    • Section 53.27: Affineness of complement of ramification locus
    • Section 53.28: Specialization maps in the smooth proper case
    • Section 53.29: 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 G-sets
    • Section 54.13: Sheafification
    • Section 54.14: Cohomology
    • Section 54.15: The fpqc topology
    • Section 54.16: Faithfully flat descent
    • Section 54.17: Quasi-coherent sheaves
    • Section 54.18: Čech cohomology
    • Section 54.19: The Čech-to-cohomology spectral sequence
    • Section 54.20: Big and small sites of schemes
    • Section 54.21: The étale topos
    • Section 54.22: Cohomology of quasi-coherent 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 Artin-Schreier 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: Affine analog of proper base change
    • Section 54.77: Cohomology of torsion sheaves on curves
    • Section 54.78: First cohomology of proper schemes
    • Section 54.79: The proper base change theorem
    • Section 54.80: Applications of proper base change
    • Section 54.81: Comparing big and small topoi
    • Section 54.82: Comparing fppf and étale topologies
    • Section 54.83: Comparing fppf and étale topologies: modules
    • Section 54.84: Comparing ph and étale topologies
    • Section 54.85: Comparing h and étale topologies
    • Section 54.86: Blow up squares and étale cohomology
    • Section 54.87: Almost blow up squares and the h topology
    • Section 54.88: Cohomology of the structure sheaf in the h topology
    • Section 54.89: The trace formula
    • Section 54.90: Frobenii
    • Section 54.91: Traces
    • Section 54.92: Why derived categories?
    • Section 54.93: Derived categories
    • Section 54.94: Filtered derived category
    • Section 54.95: Filtered derived functors
    • Section 54.96: Application of filtered complexes
    • Section 54.97: Perfectness
    • Section 54.98: Filtrations and perfect complexes
    • Section 54.99: Characterizing perfect objects
    • Section 54.100: Complexes with constructible cohomology
    • Section 54.101: Cohomology of nice complexes
    • Section 54.102: Lefschetz numbers
    • Section 54.103: Preliminaries and sorites
    • Section 54.104: Proof of the trace formula
    • Section 54.105: Applications
    • Section 54.106: On l-adic sheaves
    • Section 54.107: L-functions
    • Section 54.108: Cohomological interpretation
    • Section 54.109: List of things which we should add above
    • Section 54.110: Examples of L-functions
    • Section 54.111: Constant sheaves
    • Section 54.112: The Legendre family
    • Section 54.113: Exponential sums
    • Section 54.114: Trace formula in terms of fundamental groups
    • Section 54.115: Fundamental groups
    • Section 54.116: Profinite groups, cohomology and homology
    • Section 54.117: Cohomology of curves, revisited
    • Section 54.118: Abstract trace formula
    • Section 54.119: Automorphic forms and sheaves
    • Section 54.120: Counting points
    • Section 54.121: Precise form of Chebotarev
    • Section 54.122: How many primes decompose completely?
    • Section 54.123: 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 quasi-coherent 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: Pro-étale Cohomology
    • Section 56.1: Introduction
    • Section 56.2: Some topology
    • Section 56.3: Local isomorphisms
    • Section 56.4: Ind-Zariski algebra
    • Section 56.5: Constructing w-local affine schemes
    • Section 56.6: Identifying local rings versus ind-Zariski
    • Section 56.7: Ind-étale algebra
    • Section 56.8: Constructing ind-étale algebras
    • Section 56.9: Weakly étale versus pro-étale
    • Section 56.10: The V topology and the pro-h topology
    • Section 56.11: Constructing w-contractible covers
    • Section 56.12: The pro-étale site
    • Section 56.13: Points of the pro-étale site
    • Section 56.14: Compact generation
    • Section 56.15: Derived completion in the constant Noetherian case
    • Section 56.16: Derived completion on the pro-étale 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 pro-étale site
    • Section 56.21: Finite morphisms and pro-étale sites
    • Section 56.22: Closed immersions and pro-étale sites
    • Section 56.23: Extension by zero
    • Section 56.24: Constructible sheaves on the pro-étale site
    • Section 56.25: Constructible adic sheaves
    • Section 56.26: A suitable derived category
    • Section 56.27: Proper base change