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*}

7 Algebraic Stacks

  • Chapter 86: Algebraic Stacks
    • Section 86.1: Introduction
    • Section 86.2: Conventions
    • Section 86.3: Notation
    • Section 86.4: Representable categories fibred in groupoids
    • Section 86.5: The 2-Yoneda lemma
    • Section 86.6: Representable morphisms of categories fibred in groupoids
    • Section 86.7: Split categories fibred in groupoids
    • Section 86.8: Categories fibred in groupoids representable by algebraic spaces
    • Section 86.9: Morphisms representable by algebraic spaces
    • Section 86.10: Properties of morphisms representable by algebraic spaces
    • Section 86.11: Stacks in groupoids
    • Section 86.12: Algebraic stacks
    • Section 86.13: Algebraic stacks and algebraic spaces
    • Section 86.14: 2-Fibre products of algebraic stacks
    • Section 86.15: Algebraic stacks, overhauled
    • Section 86.16: From an algebraic stack to a presentation
    • Section 86.17: The algebraic stack associated to a smooth groupoid
    • Section 86.18: Change of big site
    • Section 86.19: Change of base scheme
  • Chapter 87: Examples of Stacks
    • Section 87.1: Introduction
    • Section 87.2: Notation
    • Section 87.3: Examples of stacks
    • Section 87.4: Quasi-coherent sheaves
    • Section 87.5: The stack of finitely generated quasi-coherent sheaves
    • Section 87.6: Finite étale covers
    • Section 87.7: Algebraic spaces
    • Section 87.8: The stack of finite type algebraic spaces
    • Section 87.9: Examples of stacks in groupoids
    • Section 87.10: The stack associated to a sheaf
    • Section 87.11: The stack in groupoids of finitely generated quasi-coherent sheaves
    • Section 87.12: The stack in groupoids of finite type algebraic spaces
    • Section 87.13: Quotient stacks
    • Section 87.14: Classifying torsors
    • Section 87.15: Quotients by group actions
    • Section 87.16: The Picard stack
    • Section 87.17: Examples of inertia stacks
    • Section 87.18: Finite Hilbert stacks
  • Chapter 88: Sheaves on Algebraic Stacks
    • Section 88.1: Introduction
    • Section 88.2: Conventions
    • Section 88.3: Presheaves
    • Section 88.4: Sheaves
    • Section 88.5: Computing pushforward
    • Section 88.6: The structure sheaf
    • Section 88.7: Sheaves of modules
    • Section 88.8: Representable categories
    • Section 88.9: Restriction
    • Section 88.10: Restriction to algebraic spaces
    • Section 88.11: Quasi-coherent modules
    • Section 88.12: Stackification and sheaves
    • Section 88.13: Quasi-coherent sheaves and presentations
    • Section 88.14: Quasi-coherent sheaves on algebraic stacks
    • Section 88.15: Cohomology
    • Section 88.16: Injective sheaves
    • Section 88.17: The Čech complex
    • Section 88.18: The relative Čech complex
    • Section 88.19: Cohomology on algebraic stacks
    • Section 88.20: Higher direct images and algebraic stacks
    • Section 88.21: Comparison
    • Section 88.22: Change of topology
  • Chapter 89: Criteria for Representability
    • Section 89.1: Introduction
    • Section 89.2: Conventions
    • Section 89.3: What we already know
    • Section 89.4: Morphisms of stacks in groupoids
    • Section 89.5: Limit preserving on objects
    • Section 89.6: Formally smooth on objects
    • Section 89.7: Surjective on objects
    • Section 89.8: Algebraic morphisms
    • Section 89.9: Spaces of sections
    • Section 89.10: Relative morphisms
    • Section 89.11: Restriction of scalars
    • Section 89.12: Finite Hilbert stacks
    • Section 89.13: The finite Hilbert stack of a point
    • Section 89.14: Finite Hilbert stacks of spaces
    • Section 89.15: LCI locus in the Hilbert stack
    • Section 89.16: Bootstrapping algebraic stacks
    • Section 89.17: Applications
    • Section 89.18: When is a quotient stack algebraic?
    • Section 89.19: Algebraic stacks in the étale topology
  • Chapter 90: Artin's axioms
    • Section 90.1: Introduction
    • Section 90.2: Conventions
    • Section 90.3: Predeformation categories
    • Section 90.4: Pushouts and stacks
    • Section 90.5: The Rim-Schlessinger condition
    • Section 90.6: Deformation categories
    • Section 90.7: Change of field
    • Section 90.8: Tangent spaces
    • Section 90.9: Formal objects
    • Section 90.10: Approximation
    • Section 90.11: Limit preserving
    • Section 90.12: Versality
    • Section 90.13: Openness of versality
    • Section 90.14: Axioms
    • Section 90.15: Axioms for functors
    • Section 90.16: Algebraic spaces
    • Section 90.17: Algebraic stacks
    • Section 90.18: Strong Rim-Schlessinger
    • Section 90.19: Strong formal effectiveness
    • Section 90.20: Infinitesimal deformations
    • Section 90.21: Obstruction theories
    • Section 90.22: Naive obstruction theories
    • Section 90.23: A dual notion
  • Chapter 91: Quot and Hilbert Spaces
    • Section 91.1: Introduction
    • Section 91.2: Conventions
    • Section 91.3: The Hom functor
    • Section 91.4: The Isom functor
    • Section 91.5: The stack of coherent sheaves
    • Section 91.6: The stack of coherent sheaves in the non-flat case
    • Section 91.7: The functor of quotients
    • Section 91.8: The Quot functor
    • Section 91.9: The Hilbert functor
    • Section 91.10: The Picard stack
    • Section 91.11: The Picard functor
    • Section 91.12: Relative morphisms
    • Section 91.13: The stack of algebraic spaces
    • Section 91.14: The stack of polarized proper schemes
    • Section 91.15: The stack of curves
    • Section 91.16: Moduli of complexes on a proper morphism
  • Chapter 92: Properties of Algebraic Stacks
    • Section 92.1: Introduction
    • Section 92.2: Conventions and abuse of language
    • Section 92.3: Properties of morphisms representable by algebraic spaces
    • Section 92.4: Points of algebraic stacks
    • Section 92.5: Surjective morphisms
    • Section 92.6: Quasi-compact algebraic stacks
    • Section 92.7: Properties of algebraic stacks defined by properties of schemes
    • Section 92.8: Monomorphisms of algebraic stacks
    • Section 92.9: Immersions of algebraic stacks
    • Section 92.10: Reduced algebraic stacks
    • Section 92.11: Residual gerbes
    • Section 92.12: Dimension of a stack
    • Section 92.13: Local irreducibility
    • Section 92.14: Finiteness conditions and points
  • Chapter 93: Morphisms of Algebraic Stacks
    • Section 93.1: Introduction
    • Section 93.2: Conventions and abuse of language
    • Section 93.3: Properties of diagonals
    • Section 93.4: Separation axioms
    • Section 93.5: Inertia stacks
    • Section 93.6: Higher diagonals
    • Section 93.7: Quasi-compact morphisms
    • Section 93.8: Noetherian algebraic stacks
    • Section 93.9: Affine morphisms
    • Section 93.10: Integral and finite morphisms
    • Section 93.11: Open morphisms
    • Section 93.12: Submersive morphisms
    • Section 93.13: Universally closed morphisms
    • Section 93.14: Universally injective morphisms
    • Section 93.15: Universal homeomorphisms
    • Section 93.16: Types of morphisms smooth local on source-and-target
    • Section 93.17: Morphisms of finite type
    • Section 93.18: Points of finite type
    • Section 93.19: Automorphism groups
    • Section 93.20: Presentations and properties of algebraic stacks
    • Section 93.21: Special presentations of algebraic stacks
    • Section 93.22: The Deligne-Mumford locus
    • Section 93.23: Quasi-finite morphisms
    • Section 93.24: Flat morphisms
    • Section 93.25: Flat at a point
    • Section 93.26: Morphisms of finite presentation
    • Section 93.27: Gerbes
    • Section 93.28: Stratification by gerbes
    • Section 93.29: The topological space of an algebraic stack
    • Section 93.30: Existence of residual gerbes
    • Section 93.31: Étale local structure
    • Section 93.32: Smooth morphisms
    • Section 93.33: Types of morphisms étale-smooth local on source-and-target
    • Section 93.34: Étale morphisms
    • Section 93.35: Unramified morphisms
    • Section 93.36: Proper morphisms
    • Section 93.37: Scheme theoretic image
    • Section 93.38: Valuative criteria
    • Section 93.39: Valuative criterion for second diagonal
    • Section 93.40: Valuative criterion for the diagonal
    • Section 93.41: Valuative criterion for universal closedness
    • Section 93.42: Valuative criterion for properness
    • Section 93.43: Local complete intersection morphisms
    • Section 93.44: Stabilizer preserving morphisms
  • Chapter 94: Limits of Algebraic Stacks
    • Section 94.1: Introduction
    • Section 94.2: Conventions
    • Section 94.3: Morphisms of finite presentation
    • Section 94.4: Descending properties
    • Section 94.5: Descending relative objects
    • Section 94.6: Finite type closed in finite presentation
  • Chapter 95: Cohomology of Algebraic Stacks
    • Section 95.1: Introduction
    • Section 95.2: Conventions and abuse of language
    • Section 95.3: Notation
    • Section 95.4: Pullback of quasi-coherent modules
    • Section 95.5: The key lemma
    • Section 95.6: Locally quasi-coherent modules
    • Section 95.7: Flat comparison maps
    • Section 95.8: Parasitic modules
    • Section 95.9: Quasi-coherent modules, I
    • Section 95.10: Pushforward of quasi-coherent modules
    • Section 95.11: The lisse-étale and the flat-fppf sites
    • Section 95.12: Quasi-coherent modules, II
  • Chapter 96: Derived Categories of Stacks
    • Section 96.1: Introduction
    • Section 96.2: Conventions, notation, and abuse of language
    • Section 96.3: The lisse-étale and the flat-fppf sites
    • Section 96.4: Derived categories of quasi-coherent modules
    • Section 96.5: Derived pushforward of quasi-coherent modules
    • Section 96.6: Derived pullback of quasi-coherent modules
  • Chapter 97: Introducing Algebraic Stacks
    • Section 97.1: Why read this?
    • Section 97.2: Preliminary
    • Section 97.3: The moduli stack of elliptic curves
    • Section 97.4: Fibre products
    • Section 97.5: The definition
    • Section 97.6: A smooth cover
    • Section 97.7: Properties of algebraic stacks
  • Chapter 98: More on Morphisms of Stacks
    • Section 98.1: Introduction
    • Section 98.2: Conventions and abuse of language
    • Section 98.3: Thickenings
    • Section 98.4: Morphisms of thickenings
    • Section 98.5: Infinitesimal deformations of algebraic stacks
    • Section 98.6: Lifting affines
    • Section 98.7: Infinitesimal deformations
    • Section 98.8: Formally smooth morphisms
    • Section 98.9: Blowing up and flatness
    • Section 98.10: Chow's lemma for algebraic stacks
    • Section 98.11: Noetherian valuative criterion
    • Section 98.12: Moduli spaces
    • Section 98.13: The Keel-Mori theorem
    • Section 98.14: Properties of moduli spaces
  • Chapter 99: The Geometry of Algebraic Stacks
    • Section 99.1: Introduction
    • Section 99.2: Versal rings
    • Section 99.3: Multiplicities of components of algebraic stacks
    • Section 99.4: Formal branches and multiplicities
    • Section 99.5: Dimension theory of algebraic stacks
    • Section 99.6: The dimension of the local ring