7 Algebraic Stacks

Chapter 93: Algebraic Stacks
 Section 93.1: Introduction
 Section 93.2: Conventions
 Section 93.3: Notation
 Section 93.4: Representable categories fibred in groupoids
 Section 93.5: The 2Yoneda lemma
 Section 93.6: Representable morphisms of categories fibred in groupoids
 Section 93.7: Split categories fibred in groupoids
 Section 93.8: Categories fibred in groupoids representable by algebraic spaces
 Section 93.9: Morphisms representable by algebraic spaces
 Section 93.10: Properties of morphisms representable by algebraic spaces
 Section 93.11: Stacks in groupoids
 Section 93.12: Algebraic stacks
 Section 93.13: Algebraic stacks and algebraic spaces
 Section 93.14: 2Fibre products of algebraic stacks
 Section 93.15: Algebraic stacks, overhauled
 Section 93.16: From an algebraic stack to a presentation
 Section 93.17: The algebraic stack associated to a smooth groupoid
 Section 93.18: Change of big site
 Section 93.19: Change of base scheme

Chapter 94: Examples of Stacks
 Section 94.1: Introduction
 Section 94.2: Notation
 Section 94.3: Examples of stacks
 Section 94.4: Quasicoherent sheaves
 Section 94.5: The stack of finitely generated quasicoherent sheaves
 Section 94.6: Finite étale covers
 Section 94.7: Algebraic spaces
 Section 94.8: The stack of finite type algebraic spaces
 Section 94.9: Examples of stacks in groupoids
 Section 94.10: The stack associated to a sheaf
 Section 94.11: The stack in groupoids of finitely generated quasicoherent sheaves
 Section 94.12: The stack in groupoids of finite type algebraic spaces
 Section 94.13: Quotient stacks
 Section 94.14: Classifying torsors
 Section 94.15: Quotients by group actions
 Section 94.16: The Picard stack
 Section 94.17: Examples of inertia stacks
 Section 94.18: Finite Hilbert stacks

Chapter 95: Sheaves on Algebraic Stacks
 Section 95.1: Introduction
 Section 95.2: Conventions
 Section 95.3: Presheaves
 Section 95.4: Sheaves
 Section 95.5: Computing pushforward
 Section 95.6: The structure sheaf
 Section 95.7: Sheaves of modules
 Section 95.8: Representable categories
 Section 95.9: Restriction
 Section 95.10: Restriction to algebraic spaces
 Section 95.11: Quasicoherent modules
 Section 95.12: Locally quasicoherent modules
 Section 95.13: Stackification and sheaves
 Section 95.14: Quasicoherent sheaves and presentations
 Section 95.15: Quasicoherent sheaves on algebraic stacks
 Section 95.16: Cohomology
 Section 95.17: Injective sheaves
 Section 95.18: The Čech complex
 Section 95.19: The relative Čech complex
 Section 95.20: Cohomology on algebraic stacks
 Section 95.21: Higher direct images and algebraic stacks
 Section 95.22: Comparison
 Section 95.23: Change of topology
 Section 95.24: Restricting to affines
 Section 95.25: Quasicoherent modules and affines
 Section 95.26: Quasicoherent objects in the derived category

Chapter 96: Criteria for Representability
 Section 96.1: Introduction
 Section 96.2: Conventions
 Section 96.3: What we already know
 Section 96.4: Morphisms of stacks in groupoids
 Section 96.5: Limit preserving on objects
 Section 96.6: Formally smooth on objects
 Section 96.7: Surjective on objects
 Section 96.8: Algebraic morphisms
 Section 96.9: Spaces of sections
 Section 96.10: Relative morphisms
 Section 96.11: Restriction of scalars
 Section 96.12: Finite Hilbert stacks
 Section 96.13: The finite Hilbert stack of a point
 Section 96.14: Finite Hilbert stacks of spaces
 Section 96.15: LCI locus in the Hilbert stack
 Section 96.16: Bootstrapping algebraic stacks
 Section 96.17: Applications
 Section 96.18: When is a quotient stack algebraic?
 Section 96.19: Algebraic stacks in the étale topology

Chapter 97: Artin's Axioms
 Section 97.1: Introduction
 Section 97.2: Conventions
 Section 97.3: Predeformation categories
 Section 97.4: Pushouts and stacks
 Section 97.5: The RimSchlessinger condition
 Section 97.6: Deformation categories
 Section 97.7: Change of field
 Section 97.8: Tangent spaces
 Section 97.9: Formal objects
 Section 97.10: Approximation
 Section 97.11: Limit preserving
 Section 97.12: Versality
 Section 97.13: Openness of versality
 Section 97.14: Axioms
 Section 97.15: Axioms for functors
 Section 97.16: Algebraic spaces
 Section 97.17: Algebraic stacks
 Section 97.18: Strong RimSchlessinger
 Section 97.19: Versality and generalizations
 Section 97.20: Strong formal effectiveness
 Section 97.21: Infinitesimal deformations
 Section 97.22: Obstruction theories
 Section 97.23: Naive obstruction theories
 Section 97.24: A dual notion
 Section 97.25: Limit preserving functors on Noetherian schemes
 Section 97.26: Algebraic spaces in the Noetherian setting
 Section 97.27: Artin's theorem on contractions

Chapter 98: Quot and Hilbert Spaces
 Section 98.1: Introduction
 Section 98.2: Conventions
 Section 98.3: The Hom functor
 Section 98.4: The Isom functor
 Section 98.5: The stack of coherent sheaves
 Section 98.6: The stack of coherent sheaves in the nonflat case
 Section 98.7: The functor of quotients
 Section 98.8: The Quot functor
 Section 98.9: The Hilbert functor
 Section 98.10: The Picard stack
 Section 98.11: The Picard functor
 Section 98.12: Relative morphisms
 Section 98.13: The stack of algebraic spaces
 Section 98.14: The stack of polarized proper schemes
 Section 98.15: The stack of curves
 Section 98.16: Moduli of complexes on a proper morphism

Chapter 99: Properties of Algebraic Stacks
 Section 99.1: Introduction
 Section 99.2: Conventions and abuse of language
 Section 99.3: Properties of morphisms representable by algebraic spaces
 Section 99.4: Points of algebraic stacks
 Section 99.5: Surjective morphisms
 Section 99.6: Quasicompact algebraic stacks
 Section 99.7: Properties of algebraic stacks defined by properties of schemes
 Section 99.8: Monomorphisms of algebraic stacks
 Section 99.9: Immersions of algebraic stacks
 Section 99.10: Reduced algebraic stacks
 Section 99.11: Residual gerbes
 Section 99.12: Dimension of a stack
 Section 99.13: Local irreducibility
 Section 99.14: Finiteness conditions and points

Chapter 100: Morphisms of Algebraic Stacks
 Section 100.1: Introduction
 Section 100.2: Conventions and abuse of language
 Section 100.3: Properties of diagonals
 Section 100.4: Separation axioms
 Section 100.5: Inertia stacks
 Section 100.6: Higher diagonals
 Section 100.7: Quasicompact morphisms
 Section 100.8: Noetherian algebraic stacks
 Section 100.9: Affine morphisms
 Section 100.10: Integral and finite morphisms
 Section 100.11: Open morphisms
 Section 100.12: Submersive morphisms
 Section 100.13: Universally closed morphisms
 Section 100.14: Universally injective morphisms
 Section 100.15: Universal homeomorphisms
 Section 100.16: Types of morphisms smooth local on sourceandtarget
 Section 100.17: Morphisms of finite type
 Section 100.18: Points of finite type
 Section 100.19: Automorphism groups
 Section 100.20: Presentations and properties of algebraic stacks
 Section 100.21: Special presentations of algebraic stacks
 Section 100.22: The DeligneMumford locus
 Section 100.23: Locally quasifinite morphisms
 Section 100.24: Quasifinite morphisms
 Section 100.25: Flat morphisms
 Section 100.26: Flat at a point
 Section 100.27: Morphisms of finite presentation
 Section 100.28: Gerbes
 Section 100.29: Stratification by gerbes
 Section 100.30: The topological space of an algebraic stack
 Section 100.31: Existence of residual gerbes
 Section 100.32: Étale local structure
 Section 100.33: Smooth morphisms
 Section 100.34: Types of morphisms étalesmooth local on sourceandtarget
 Section 100.35: Étale morphisms
 Section 100.36: Unramified morphisms
 Section 100.37: Proper morphisms
 Section 100.38: Scheme theoretic image
 Section 100.39: Valuative criteria
 Section 100.40: Valuative criterion for second diagonal
 Section 100.41: Valuative criterion for the diagonal
 Section 100.42: Valuative criterion for universal closedness
 Section 100.43: Valuative criterion for properness
 Section 100.44: Local complete intersection morphisms
 Section 100.45: Stabilizer preserving morphisms
 Section 100.46: Normalization
 Section 100.47: Points and specializations
 Section 100.48: Decent algebraic stacks
 Section 100.49: Points on decent stacks
 Section 100.50: Integral algebraic stacks
 Section 100.51: Residual gerbes
 Chapter 101: Limits of Algebraic Stacks

Chapter 102: Cohomology of Algebraic Stacks
 Section 102.1: Introduction
 Section 102.2: Conventions and abuse of language
 Section 102.3: Notation
 Section 102.4: Pullback of quasicoherent modules
 Section 102.5: Higher direct images of types of modules
 Section 102.6: Locally quasicoherent modules
 Section 102.7: Flat comparison maps
 Section 102.8: Locally quasicoherent modules with the flat base change property
 Section 102.9: Parasitic modules
 Section 102.10: Quasicoherent modules
 Section 102.11: Pushforward of quasicoherent modules
 Section 102.12: Further remarks on quasicoherent modules
 Section 102.13: Colimits and cohomology
 Section 102.14: The lisseétale and the flatfppf sites
 Section 102.15: Functoriality of the lisseétale and flatfppf sites
 Section 102.16: Quasicoherent modules and the lisseétale and flatfppf sites
 Section 102.17: Coherent sheaves on locally Noetherian stacks
 Section 102.18: Coherent sheaves on Noetherian stacks

Chapter 103: Derived Categories of Stacks
 Section 103.1: Introduction
 Section 103.2: Conventions, notation, and abuse of language
 Section 103.3: The lisseétale and the flatfppf sites
 Section 103.4: Cohomology and the lisseétale and flatfppf sites
 Section 103.5: Derived categories of quasicoherent modules
 Section 103.6: Derived pushforward of quasicoherent modules
 Section 103.7: Derived pullback of quasicoherent modules
 Section 103.8: Quasicoherent objects in the derived category
 Chapter 104: Introducing Algebraic Stacks

Chapter 105: More on Morphisms of Stacks
 Section 105.1: Introduction
 Section 105.2: Conventions and abuse of language
 Section 105.3: Thickenings
 Section 105.4: Morphisms of thickenings
 Section 105.5: Infinitesimal deformations of algebraic stacks
 Section 105.6: Lifting affines
 Section 105.7: Infinitesimal deformations
 Section 105.8: Formally smooth morphisms
 Section 105.9: Blowing up and flatness
 Section 105.10: Chow's lemma for algebraic stacks
 Section 105.11: Noetherian valuative criterion
 Section 105.12: Moduli spaces
 Section 105.13: The KeelMori theorem
 Section 105.14: Properties of moduli spaces
 Section 105.15: Stacks and fpqc coverings
 Section 105.16: Tensor functors
 Chapter 106: The Geometry of Algebraic Stacks