The Stacks project

6 Deformation Theory

  • Chapter 88: Formal Deformation Theory
    • Section 88.1: Introduction
    • Section 88.2: Notation and Conventions
    • Section 88.3: The base category
    • Section 88.4: The completed base category
    • Section 88.5: Categories cofibered in groupoids
    • Section 88.6: Prorepresentable functors and predeformation categories
    • Section 88.7: Formal objects and completion categories
    • Section 88.8: Smooth morphisms
    • Section 88.9: Smooth or unobstructed categories
    • Section 88.10: Schlessinger's conditions
    • Section 88.11: Tangent spaces of functors
    • Section 88.12: Tangent spaces of predeformation categories
    • Section 88.13: Versal formal objects
    • Section 88.14: Minimal versal formal objects
    • Section 88.15: Miniversal formal objects and tangent spaces
    • Section 88.16: Rim-Schlessinger conditions and deformation categories
    • Section 88.17: Lifts of objects
    • Section 88.18: Schlessinger's theorem on prorepresentable functors
    • Section 88.19: Infinitesimal automorphisms
    • Section 88.20: Applications
    • Section 88.21: Groupoids in functors on an arbitrary category
    • Section 88.22: Groupoids in functors on the base category
    • Section 88.23: Smooth groupoids in functors on the base category
    • Section 88.24: Deformation categories as quotients of groupoids in functors
    • Section 88.25: Presentations of categories cofibered in groupoids
    • Section 88.26: Presentations of deformation categories
    • Section 88.27: Remarks regarding minimality
    • Section 88.28: Uniqueness of versal rings
    • Section 88.29: Change of residue field
  • Chapter 89: Deformation Theory
    • Section 89.1: Introduction
    • Section 89.2: Deformations of rings and the naive cotangent complex
    • Section 89.3: Thickenings of ringed spaces
    • Section 89.4: Modules on first order thickenings of ringed spaces
    • Section 89.5: Infinitesimal deformations of modules on ringed spaces
    • Section 89.6: Application to flat modules on flat thickenings of ringed spaces
    • Section 89.7: Deformations of ringed spaces and the naive cotangent complex
    • Section 89.8: Deformations of schemes
    • Section 89.9: Thickenings of ringed topoi
    • Section 89.10: Modules on first order thickenings of ringed topoi
    • Section 89.11: Infinitesimal deformations of modules on ringed topoi
    • Section 89.12: Application to flat modules on flat thickenings of ringed topoi
    • Section 89.13: Deformations of ringed topoi and the naive cotangent complex
    • Section 89.14: Deformations of algebraic spaces
    • Section 89.15: Deformations of complexes
    • Section 89.16: Deformations of complexes on ringed topoi
  • Chapter 90: The Cotangent Complex
    • Section 90.1: Introduction
    • Section 90.2: Advice for the reader
    • Section 90.3: The cotangent complex of a ring map
    • Section 90.4: Simplicial resolutions and derived lower shriek
    • Section 90.5: Constructing a resolution
    • Section 90.6: Functoriality
    • Section 90.7: The fundamental triangle
    • Section 90.8: Localization and ├ętale ring maps
    • Section 90.9: Smooth ring maps
    • Section 90.10: Comparison with the naive cotangent complex
    • Section 90.11: A spectral sequence of Quillen
    • Section 90.12: Comparison with Lichtenbaum-Schlessinger
    • Section 90.13: The cotangent complex of a local complete intersection
    • Section 90.14: Tensor products and the cotangent complex
    • Section 90.15: Deformations of ring maps and the cotangent complex
    • Section 90.16: The Atiyah class of a module
    • Section 90.17: The cotangent complex
    • Section 90.18: The Atiyah class of a sheaf of modules
    • Section 90.19: The cotangent complex of a morphism of ringed spaces
    • Section 90.20: Deformations of ringed spaces and the cotangent complex
    • Section 90.21: The cotangent complex of a morphism of ringed topoi
    • Section 90.22: Deformations of ringed topoi and the cotangent complex
    • Section 90.23: The cotangent complex of a morphism of schemes
    • Section 90.24: The cotangent complex of a scheme over a ring
    • Section 90.25: The cotangent complex of a morphism of algebraic spaces
    • Section 90.26: The cotangent complex of an algebraic space over a ring
    • Section 90.27: Fibre products of algebraic spaces and the cotangent complex
  • Chapter 91: Deformation Problems
    • Section 91.1: Introduction
    • Section 91.2: Examples of deformation problems
    • Section 91.3: General outline
    • Section 91.4: Finite projective modules
    • Section 91.5: Representations of a group
    • Section 91.6: Continuous representations
    • Section 91.7: Graded algebras
    • Section 91.8: Rings
    • Section 91.9: Schemes
    • Section 91.10: Morphisms of Schemes
    • Section 91.11: Algebraic spaces
    • Section 91.12: Deformations of completions
    • Section 91.13: Deformations of localizations
    • Section 91.14: Deformations of henselizations
    • Section 91.15: Application to isolated singularities
    • Section 91.16: Unobstructed deformation problems
    • Section 91.17: Smoothings