The Stacks project

7 Algebraic Stacks

  • Chapter 92: Algebraic Stacks
    • Section 92.1: Introduction
    • Section 92.2: Conventions
    • Section 92.3: Notation
    • Section 92.4: Representable categories fibred in groupoids
    • Section 92.5: The 2-Yoneda lemma
    • Section 92.6: Representable morphisms of categories fibred in groupoids
    • Section 92.7: Split categories fibred in groupoids
    • Section 92.8: Categories fibred in groupoids representable by algebraic spaces
    • Section 92.9: Morphisms representable by algebraic spaces
    • Section 92.10: Properties of morphisms representable by algebraic spaces
    • Section 92.11: Stacks in groupoids
    • Section 92.12: Algebraic stacks
    • Section 92.13: Algebraic stacks and algebraic spaces
    • Section 92.14: 2-Fibre products of algebraic stacks
    • Section 92.15: Algebraic stacks, overhauled
    • Section 92.16: From an algebraic stack to a presentation
    • Section 92.17: The algebraic stack associated to a smooth groupoid
    • Section 92.18: Change of big site
    • Section 92.19: Change of base scheme
  • Chapter 93: Examples of Stacks
    • Section 93.1: Introduction
    • Section 93.2: Notation
    • Section 93.3: Examples of stacks
    • Section 93.4: Quasi-coherent sheaves
    • Section 93.5: The stack of finitely generated quasi-coherent sheaves
    • Section 93.6: Finite étale covers
    • Section 93.7: Algebraic spaces
    • Section 93.8: The stack of finite type algebraic spaces
    • Section 93.9: Examples of stacks in groupoids
    • Section 93.10: The stack associated to a sheaf
    • Section 93.11: The stack in groupoids of finitely generated quasi-coherent sheaves
    • Section 93.12: The stack in groupoids of finite type algebraic spaces
    • Section 93.13: Quotient stacks
    • Section 93.14: Classifying torsors
    • Section 93.15: Quotients by group actions
    • Section 93.16: The Picard stack
    • Section 93.17: Examples of inertia stacks
    • Section 93.18: Finite Hilbert stacks
  • Chapter 94: Sheaves on Algebraic Stacks
    • Section 94.1: Introduction
    • Section 94.2: Conventions
    • Section 94.3: Presheaves
    • Section 94.4: Sheaves
    • Section 94.5: Computing pushforward
    • Section 94.6: The structure sheaf
    • Section 94.7: Sheaves of modules
    • Section 94.8: Representable categories
    • Section 94.9: Restriction
    • Section 94.10: Restriction to algebraic spaces
    • Section 94.11: Quasi-coherent modules
    • Section 94.12: Stackification and sheaves
    • Section 94.13: Quasi-coherent sheaves and presentations
    • Section 94.14: Quasi-coherent sheaves on algebraic stacks
    • Section 94.15: Cohomology
    • Section 94.16: Injective sheaves
    • Section 94.17: The Čech complex
    • Section 94.18: The relative Čech complex
    • Section 94.19: Cohomology on algebraic stacks
    • Section 94.20: Higher direct images and algebraic stacks
    • Section 94.21: Comparison
    • Section 94.22: Change of topology
  • Chapter 95: Criteria for Representability
    • Section 95.1: Introduction
    • Section 95.2: Conventions
    • Section 95.3: What we already know
    • Section 95.4: Morphisms of stacks in groupoids
    • Section 95.5: Limit preserving on objects
    • Section 95.6: Formally smooth on objects
    • Section 95.7: Surjective on objects
    • Section 95.8: Algebraic morphisms
    • Section 95.9: Spaces of sections
    • Section 95.10: Relative morphisms
    • Section 95.11: Restriction of scalars
    • Section 95.12: Finite Hilbert stacks
    • Section 95.13: The finite Hilbert stack of a point
    • Section 95.14: Finite Hilbert stacks of spaces
    • Section 95.15: LCI locus in the Hilbert stack
    • Section 95.16: Bootstrapping algebraic stacks
    • Section 95.17: Applications
    • Section 95.18: When is a quotient stack algebraic?
    • Section 95.19: Algebraic stacks in the étale topology
  • Chapter 96: Artin's Axioms
    • Section 96.1: Introduction
    • Section 96.2: Conventions
    • Section 96.3: Predeformation categories
    • Section 96.4: Pushouts and stacks
    • Section 96.5: The Rim-Schlessinger condition
    • Section 96.6: Deformation categories
    • Section 96.7: Change of field
    • Section 96.8: Tangent spaces
    • Section 96.9: Formal objects
    • Section 96.10: Approximation
    • Section 96.11: Limit preserving
    • Section 96.12: Versality
    • Section 96.13: Openness of versality
    • Section 96.14: Axioms
    • Section 96.15: Axioms for functors
    • Section 96.16: Algebraic spaces
    • Section 96.17: Algebraic stacks
    • Section 96.18: Strong Rim-Schlessinger
    • Section 96.19: Strong formal effectiveness
    • Section 96.20: Infinitesimal deformations
    • Section 96.21: Obstruction theories
    • Section 96.22: Naive obstruction theories
    • Section 96.23: A dual notion
  • Chapter 97: Quot and Hilbert Spaces
    • Section 97.1: Introduction
    • Section 97.2: Conventions
    • Section 97.3: The Hom functor
    • Section 97.4: The Isom functor
    • Section 97.5: The stack of coherent sheaves
    • Section 97.6: The stack of coherent sheaves in the non-flat case
    • Section 97.7: The functor of quotients
    • Section 97.8: The Quot functor
    • Section 97.9: The Hilbert functor
    • Section 97.10: The Picard stack
    • Section 97.11: The Picard functor
    • Section 97.12: Relative morphisms
    • Section 97.13: The stack of algebraic spaces
    • Section 97.14: The stack of polarized proper schemes
    • Section 97.15: The stack of curves
    • Section 97.16: Moduli of complexes on a proper morphism
  • Chapter 98: Properties of Algebraic Stacks
    • Section 98.1: Introduction
    • Section 98.2: Conventions and abuse of language
    • Section 98.3: Properties of morphisms representable by algebraic spaces
    • Section 98.4: Points of algebraic stacks
    • Section 98.5: Surjective morphisms
    • Section 98.6: Quasi-compact algebraic stacks
    • Section 98.7: Properties of algebraic stacks defined by properties of schemes
    • Section 98.8: Monomorphisms of algebraic stacks
    • Section 98.9: Immersions of algebraic stacks
    • Section 98.10: Reduced algebraic stacks
    • Section 98.11: Residual gerbes
    • Section 98.12: Dimension of a stack
    • Section 98.13: Local irreducibility
    • Section 98.14: Finiteness conditions and points
  • Chapter 99: Morphisms of Algebraic Stacks
    • Section 99.1: Introduction
    • Section 99.2: Conventions and abuse of language
    • Section 99.3: Properties of diagonals
    • Section 99.4: Separation axioms
    • Section 99.5: Inertia stacks
    • Section 99.6: Higher diagonals
    • Section 99.7: Quasi-compact morphisms
    • Section 99.8: Noetherian algebraic stacks
    • Section 99.9: Affine morphisms
    • Section 99.10: Integral and finite morphisms
    • Section 99.11: Open morphisms
    • Section 99.12: Submersive morphisms
    • Section 99.13: Universally closed morphisms
    • Section 99.14: Universally injective morphisms
    • Section 99.15: Universal homeomorphisms
    • Section 99.16: Types of morphisms smooth local on source-and-target
    • Section 99.17: Morphisms of finite type
    • Section 99.18: Points of finite type
    • Section 99.19: Automorphism groups
    • Section 99.20: Presentations and properties of algebraic stacks
    • Section 99.21: Special presentations of algebraic stacks
    • Section 99.22: The Deligne-Mumford locus
    • Section 99.23: Quasi-finite morphisms
    • Section 99.24: Flat morphisms
    • Section 99.25: Flat at a point
    • Section 99.26: Morphisms of finite presentation
    • Section 99.27: Gerbes
    • Section 99.28: Stratification by gerbes
    • Section 99.29: The topological space of an algebraic stack
    • Section 99.30: Existence of residual gerbes
    • Section 99.31: Étale local structure
    • Section 99.32: Smooth morphisms
    • Section 99.33: Types of morphisms étale-smooth local on source-and-target
    • Section 99.34: Étale morphisms
    • Section 99.35: Unramified morphisms
    • Section 99.36: Proper morphisms
    • Section 99.37: Scheme theoretic image
    • Section 99.38: Valuative criteria
    • Section 99.39: Valuative criterion for second diagonal
    • Section 99.40: Valuative criterion for the diagonal
    • Section 99.41: Valuative criterion for universal closedness
    • Section 99.42: Valuative criterion for properness
    • Section 99.43: Local complete intersection morphisms
    • Section 99.44: Stabilizer preserving morphisms
  • Chapter 100: Limits of Algebraic Stacks
    • Section 100.1: Introduction
    • Section 100.2: Conventions
    • Section 100.3: Morphisms of finite presentation
    • Section 100.4: Descending properties
    • Section 100.5: Descending relative objects
    • Section 100.6: Finite type closed in finite presentation
  • Chapter 101: Cohomology of Algebraic Stacks
    • Section 101.1: Introduction
    • Section 101.2: Conventions and abuse of language
    • Section 101.3: Notation
    • Section 101.4: Pullback of quasi-coherent modules
    • Section 101.5: The key lemma
    • Section 101.6: Locally quasi-coherent modules
    • Section 101.7: Flat comparison maps
    • Section 101.8: Parasitic modules
    • Section 101.9: Quasi-coherent modules, I
    • Section 101.10: Pushforward of quasi-coherent modules
    • Section 101.11: The lisse-étale and the flat-fppf sites
    • Section 101.12: Quasi-coherent modules, II
  • Chapter 102: Derived Categories of Stacks
    • Section 102.1: Introduction
    • Section 102.2: Conventions, notation, and abuse of language
    • Section 102.3: The lisse-étale and the flat-fppf sites
    • Section 102.4: Derived categories of quasi-coherent modules
    • Section 102.5: Derived pushforward of quasi-coherent modules
    • Section 102.6: Derived pullback of quasi-coherent modules
  • Chapter 103: Introducing Algebraic Stacks
    • Section 103.1: Why read this?
    • Section 103.2: Preliminary
    • Section 103.3: The moduli stack of elliptic curves
    • Section 103.4: Fibre products
    • Section 103.5: The definition
    • Section 103.6: A smooth cover
    • Section 103.7: Properties of algebraic stacks
  • Chapter 104: More on Morphisms of Stacks
    • Section 104.1: Introduction
    • Section 104.2: Conventions and abuse of language
    • Section 104.3: Thickenings
    • Section 104.4: Morphisms of thickenings
    • Section 104.5: Infinitesimal deformations of algebraic stacks
    • Section 104.6: Lifting affines
    • Section 104.7: Infinitesimal deformations
    • Section 104.8: Formally smooth morphisms
    • Section 104.9: Blowing up and flatness
    • Section 104.10: Chow's lemma for algebraic stacks
    • Section 104.11: Noetherian valuative criterion
    • Section 104.12: Moduli spaces
    • Section 104.13: The Keel-Mori theorem
    • Section 104.14: Properties of moduli spaces
  • Chapter 105: The Geometry of Algebraic Stacks
    • Section 105.1: Introduction
    • Section 105.2: Versal rings
    • Section 105.3: Multiplicities of components of algebraic stacks
    • Section 105.4: Formal branches and multiplicities
    • Section 105.5: Dimension theory of algebraic stacks
    • Section 105.6: The dimension of the local ring