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 2Yoneda 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: 2Fibre 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: Quasicoherent sheaves
 Section 93.5: The stack of finitely generated quasicoherent 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 quasicoherent 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: Quasicoherent modules
 Section 94.12: Stackification and sheaves
 Section 94.13: Quasicoherent sheaves and presentations
 Section 94.14: Quasicoherent 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 RimSchlessinger 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 RimSchlessinger
 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 nonflat 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: Quasicompact 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: Quasicompact 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 sourceandtarget
 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 DeligneMumford locus
 Section 99.23: Quasifinite 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 étalesmooth local on sourceandtarget
 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

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 quasicoherent modules
 Section 101.5: The key lemma
 Section 101.6: Locally quasicoherent modules
 Section 101.7: Flat comparison maps
 Section 101.8: Parasitic modules
 Section 101.9: Quasicoherent modules, I
 Section 101.10: Pushforward of quasicoherent modules
 Section 101.11: The lisseétale and the flatfppf sites
 Section 101.12: Quasicoherent 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 lisseétale and the flatfppf sites
 Section 102.4: Derived categories of quasicoherent modules
 Section 102.5: Derived pushforward of quasicoherent modules
 Section 102.6: Derived pullback of quasicoherent modules
 Chapter 103: Introducing 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 KeelMori theorem
 Section 104.14: Properties of moduli spaces
 Chapter 105: The Geometry of Algebraic Stacks