7 Algebraic Stacks

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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