The Stacks project

7 Algebraic Stacks

  • Chapter 94: Algebraic Stacks
    • Section 94.1: Introduction
    • Section 94.2: Conventions
    • Section 94.3: Notation
    • Section 94.4: Representable categories fibred in groupoids
    • Section 94.5: The 2-Yoneda lemma
    • Section 94.6: Representable morphisms of categories fibred in groupoids
    • Section 94.7: Split categories fibred in groupoids
    • Section 94.8: Categories fibred in groupoids representable by algebraic spaces
    • Section 94.9: Morphisms representable by algebraic spaces
    • Section 94.10: Properties of morphisms representable by algebraic spaces
    • Section 94.11: Stacks in groupoids
    • Section 94.12: Algebraic stacks
    • Section 94.13: Algebraic stacks and algebraic spaces
    • Section 94.14: 2-Fibre products of algebraic stacks
    • Section 94.15: Algebraic stacks, overhauled
    • Section 94.16: From an algebraic stack to a presentation
    • Section 94.17: The algebraic stack associated to a smooth groupoid
    • Section 94.18: Change of big site
    • Section 94.19: Change of base scheme
  • Chapter 95: Examples of Stacks
    • Section 95.1: Introduction
    • Section 95.2: Notation
    • Section 95.3: Examples of stacks
    • Section 95.4: Quasi-coherent sheaves
    • Section 95.5: The stack of finitely generated quasi-coherent sheaves
    • Section 95.6: Finite étale covers
    • Section 95.7: Algebraic spaces
    • Section 95.8: The stack of finite type algebraic spaces
    • Section 95.9: Examples of stacks in groupoids
    • Section 95.10: The stack associated to a sheaf
    • Section 95.11: The stack in groupoids of finitely generated quasi-coherent sheaves
    • Section 95.12: The stack in groupoids of finite type algebraic spaces
    • Section 95.13: Quotient stacks
    • Section 95.14: Classifying torsors
    • Section 95.15: Quotients by group actions
    • Section 95.16: The Picard stack
    • Section 95.17: Examples of inertia stacks
    • Section 95.18: Finite Hilbert stacks
  • Chapter 96: Sheaves on Algebraic Stacks
    • Section 96.1: Introduction
    • Section 96.2: Conventions
    • Section 96.3: Presheaves
    • Section 96.4: Sheaves
    • Section 96.5: Computing pushforward
    • Section 96.6: The structure sheaf
    • Section 96.7: Sheaves of modules
    • Section 96.8: Representable categories
    • Section 96.9: Restriction
    • Section 96.10: Restriction to algebraic spaces
    • Section 96.11: Quasi-coherent modules
    • Section 96.12: Locally quasi-coherent modules
    • Section 96.13: Stackification and sheaves
    • Section 96.14: Quasi-coherent sheaves and presentations
    • Section 96.15: Quasi-coherent sheaves on algebraic stacks
    • Section 96.16: Cohomology
    • Section 96.17: Injective sheaves
    • Section 96.18: The Čech complex
    • Section 96.19: The relative Čech complex
    • Section 96.20: Cohomology on algebraic stacks
    • Section 96.21: Higher direct images and algebraic stacks
    • Section 96.22: Comparison
    • Section 96.23: Change of topology
    • Section 96.24: Restricting to affines
    • Section 96.25: Quasi-coherent modules and affines
    • Section 96.26: Quasi-coherent objects in the derived category
  • Chapter 97: Criteria for Representability
    • Section 97.1: Introduction
    • Section 97.2: Conventions
    • Section 97.3: What we already know
    • Section 97.4: Morphisms of stacks in groupoids
    • Section 97.5: Limit preserving on objects
    • Section 97.6: Formally smooth on objects
    • Section 97.7: Surjective on objects
    • Section 97.8: Algebraic morphisms
    • Section 97.9: Spaces of sections
    • Section 97.10: Relative morphisms
    • Section 97.11: Restriction of scalars
    • Section 97.12: Finite Hilbert stacks
    • Section 97.13: The finite Hilbert stack of a point
    • Section 97.14: Finite Hilbert stacks of spaces
    • Section 97.15: LCI locus in the Hilbert stack
    • Section 97.16: Bootstrapping algebraic stacks
    • Section 97.17: Applications
    • Section 97.18: When is a quotient stack algebraic?
    • Section 97.19: Algebraic stacks in the étale topology
  • Chapter 98: Artin's Axioms
    • Section 98.1: Introduction
    • Section 98.2: Conventions
    • Section 98.3: Predeformation categories
    • Section 98.4: Pushouts and stacks
    • Section 98.5: The Rim-Schlessinger condition
    • Section 98.6: Deformation categories
    • Section 98.7: Change of field
    • Section 98.8: Tangent spaces
    • Section 98.9: Formal objects
    • Section 98.10: Approximation
    • Section 98.11: Limit preserving
    • Section 98.12: Versality
    • Section 98.13: Openness of versality
    • Section 98.14: Axioms
    • Section 98.15: Axioms for functors
    • Section 98.16: Algebraic spaces
    • Section 98.17: Algebraic stacks
    • Section 98.18: Strong Rim-Schlessinger
    • Section 98.19: Versality and generalizations
    • Section 98.20: Strong formal effectiveness
    • Section 98.21: Infinitesimal deformations
    • Section 98.22: Obstruction theories
    • Section 98.23: Naive obstruction theories
    • Section 98.24: A dual notion
    • Section 98.25: Limit preserving functors on Noetherian schemes
    • Section 98.26: Algebraic spaces in the Noetherian setting
    • Section 98.27: Artin's theorem on contractions
  • Chapter 99: Quot and Hilbert Spaces
    • Section 99.1: Introduction
    • Section 99.2: Conventions
    • Section 99.3: The Hom functor
    • Section 99.4: The Isom functor
    • Section 99.5: The stack of coherent sheaves
    • Section 99.6: The stack of coherent sheaves in the non-flat case
    • Section 99.7: The functor of quotients
    • Section 99.8: The Quot functor
    • Section 99.9: The Hilbert functor
    • Section 99.10: The Picard stack
    • Section 99.11: The Picard functor
    • Section 99.12: Relative morphisms
    • Section 99.13: The stack of algebraic spaces
    • Section 99.14: The stack of polarized proper schemes
    • Section 99.15: The stack of curves
    • Section 99.16: Moduli of complexes on a proper morphism
  • Chapter 100: Properties of Algebraic Stacks
    • Section 100.1: Introduction
    • Section 100.2: Conventions and abuse of language
    • Section 100.3: Properties of morphisms representable by algebraic spaces
    • Section 100.4: Points of algebraic stacks
    • Section 100.5: Surjective morphisms
    • Section 100.6: Quasi-compact algebraic stacks
    • Section 100.7: Properties of algebraic stacks defined by properties of schemes
    • Section 100.8: Monomorphisms of algebraic stacks
    • Section 100.9: Immersions of algebraic stacks
    • Section 100.10: Reduced algebraic stacks
    • Section 100.11: Residual gerbes
    • Section 100.12: Dimension of a stack
    • Section 100.13: Local irreducibility
    • Section 100.14: Finiteness conditions and points
  • Chapter 101: Morphisms of Algebraic Stacks
    • Section 101.1: Introduction
    • Section 101.2: Conventions and abuse of language
    • Section 101.3: Properties of diagonals
    • Section 101.4: Separation axioms
    • Section 101.5: Inertia stacks
    • Section 101.6: Higher diagonals
    • Section 101.7: Quasi-compact morphisms
    • Section 101.8: Noetherian algebraic stacks
    • Section 101.9: Affine morphisms
    • Section 101.10: Integral and finite morphisms
    • Section 101.11: Open morphisms
    • Section 101.12: Submersive morphisms
    • Section 101.13: Universally closed morphisms
    • Section 101.14: Universally injective morphisms
    • Section 101.15: Universal homeomorphisms
    • Section 101.16: Types of morphisms smooth local on source-and-target
    • Section 101.17: Morphisms of finite type
    • Section 101.18: Points of finite type
    • Section 101.19: Automorphism groups
    • Section 101.20: Presentations and properties of algebraic stacks
    • Section 101.21: Special presentations of algebraic stacks
    • Section 101.22: The Deligne-Mumford locus
    • Section 101.23: Locally quasi-finite morphisms
    • Section 101.24: Quasi-finite morphisms
    • Section 101.25: Flat morphisms
    • Section 101.26: Flat at a point
    • Section 101.27: Morphisms of finite presentation
    • Section 101.28: Gerbes
    • Section 101.29: Stratification by gerbes
    • Section 101.30: The topological space of an algebraic stack
    • Section 101.31: Existence of residual gerbes
    • Section 101.32: Étale local structure
    • Section 101.33: Smooth morphisms
    • Section 101.34: Types of morphisms étale-smooth local on source-and-target
    • Section 101.35: Étale morphisms
    • Section 101.36: Unramified morphisms
    • Section 101.37: Proper morphisms
    • Section 101.38: Scheme theoretic image
    • Section 101.39: Valuative criteria
    • Section 101.40: Valuative criterion for second diagonal
    • Section 101.41: Valuative criterion for the diagonal
    • Section 101.42: Valuative criterion for universal closedness
    • Section 101.43: Valuative criterion for properness
    • Section 101.44: Local complete intersection morphisms
    • Section 101.45: Stabilizer preserving morphisms
    • Section 101.46: Normalization
    • Section 101.47: Points and specializations
    • Section 101.48: Decent algebraic stacks
    • Section 101.49: Points on decent stacks
    • Section 101.50: Integral algebraic stacks
    • Section 101.51: Residual gerbes
  • Chapter 102: Limits of Algebraic Stacks
    • Section 102.1: Introduction
    • Section 102.2: Conventions
    • Section 102.3: Morphisms of finite presentation
    • Section 102.4: Descending properties
    • Section 102.5: Descending relative objects
    • Section 102.6: Finite type closed in finite presentation
    • Section 102.7: Universally closed morphisms
  • Chapter 103: Cohomology of Algebraic Stacks
    • Section 103.1: Introduction
    • Section 103.2: Conventions and abuse of language
    • Section 103.3: Notation
    • Section 103.4: Pullback of quasi-coherent modules
    • Section 103.5: Higher direct images of types of modules
    • Section 103.6: Locally quasi-coherent modules
    • Section 103.7: Flat comparison maps
    • Section 103.8: Locally quasi-coherent modules with the flat base change property
    • Section 103.9: Parasitic modules
    • Section 103.10: Quasi-coherent modules
    • Section 103.11: Pushforward of quasi-coherent modules
    • Section 103.12: Further remarks on quasi-coherent modules
    • Section 103.13: Colimits and cohomology
    • Section 103.14: The lisse-étale and the flat-fppf sites
    • Section 103.15: Functoriality of the lisse-étale and flat-fppf sites
    • Section 103.16: Quasi-coherent modules and the lisse-étale and flat-fppf sites
    • Section 103.17: Coherent sheaves on locally Noetherian stacks
    • Section 103.18: Coherent sheaves on Noetherian stacks
  • Chapter 104: Derived Categories of Stacks
    • Section 104.1: Introduction
    • Section 104.2: Conventions, notation, and abuse of language
    • Section 104.3: The lisse-étale and the flat-fppf sites
    • Section 104.4: Cohomology and the lisse-étale and flat-fppf sites
    • Section 104.5: Derived categories of quasi-coherent modules
    • Section 104.6: Derived pushforward of quasi-coherent modules
    • Section 104.7: Derived pullback of quasi-coherent modules
    • Section 104.8: Quasi-coherent objects in the derived category
  • Chapter 105: Introducing Algebraic Stacks
    • Section 105.1: Why read this?
    • Section 105.2: Preliminary
    • Section 105.3: The moduli stack of elliptic curves
    • Section 105.4: Fibre products
    • Section 105.5: The definition
    • Section 105.6: A smooth cover
    • Section 105.7: Properties of algebraic stacks
  • Chapter 106: More on Morphisms of Stacks
    • Section 106.1: Introduction
    • Section 106.2: Conventions and abuse of language
    • Section 106.3: Thickenings
    • Section 106.4: Morphisms of thickenings
    • Section 106.5: Infinitesimal deformations of algebraic stacks
    • Section 106.6: Lifting affines
    • Section 106.7: Infinitesimal deformations
    • Section 106.8: Formally smooth morphisms
    • Section 106.9: Blowing up and flatness
    • Section 106.10: Chow's lemma for algebraic stacks
    • Section 106.11: Noetherian valuative criterion
    • Section 106.12: Moduli spaces
    • Section 106.13: The Keel-Mori theorem
    • Section 106.14: Properties of moduli spaces
    • Section 106.15: Stacks and fpqc coverings
    • Section 106.16: Tensor functors
  • Chapter 107: The Geometry of Algebraic Stacks
    • Section 107.1: Introduction
    • Section 107.2: Versal rings
    • Section 107.3: Multiplicities of components of algebraic stacks
    • Section 107.4: Formal branches and multiplicities
    • Section 107.5: Dimension theory of algebraic stacks
    • Section 107.6: The dimension of the local ring