The Stacks project

6 Deformation Theory

  • Chapter 90: Formal Deformation Theory
    • Section 90.1: Introduction
    • Section 90.2: Notation and Conventions
    • Section 90.3: The base category
    • Section 90.4: The completed base category
    • Section 90.5: Categories cofibered in groupoids
    • Section 90.6: Prorepresentable functors and predeformation categories
    • Section 90.7: Formal objects and completion categories
    • Section 90.8: Smooth morphisms
    • Section 90.9: Smooth or unobstructed categories
    • Section 90.10: Schlessinger's conditions
    • Section 90.11: Tangent spaces of functors
    • Section 90.12: Tangent spaces of predeformation categories
    • Section 90.13: Versal formal objects
    • Section 90.14: Minimal versal formal objects
    • Section 90.15: Miniversal formal objects and tangent spaces
    • Section 90.16: Rim-Schlessinger conditions and deformation categories
    • Section 90.17: Lifts of objects
    • Section 90.18: Schlessinger's theorem on prorepresentable functors
    • Section 90.19: Infinitesimal automorphisms
    • Section 90.20: Applications
    • Section 90.21: Groupoids in functors on an arbitrary category
    • Section 90.22: Groupoids in functors on the base category
    • Section 90.23: Smooth groupoids in functors on the base category
    • Section 90.24: Deformation categories as quotients of groupoids in functors
    • Section 90.25: Presentations of categories cofibered in groupoids
    • Section 90.26: Presentations of deformation categories
    • Section 90.27: Remarks regarding minimality
    • Section 90.28: Uniqueness of versal rings
    • Section 90.29: Change of residue field
  • Chapter 91: Deformation Theory
    • Section 91.1: Introduction
    • Section 91.2: Deformations of rings and the naive cotangent complex
    • Section 91.3: Thickenings of ringed spaces
    • Section 91.4: Modules on first order thickenings of ringed spaces
    • Section 91.5: Infinitesimal deformations of modules on ringed spaces
    • Section 91.6: Application to flat modules on flat thickenings of ringed spaces
    • Section 91.7: Deformations of ringed spaces and the naive cotangent complex
    • Section 91.8: Deformations of schemes
    • Section 91.9: Thickenings of ringed topoi
    • Section 91.10: Modules on first order thickenings of ringed topoi
    • Section 91.11: Infinitesimal deformations of modules on ringed topoi
    • Section 91.12: Application to flat modules on flat thickenings of ringed topoi
    • Section 91.13: Deformations of ringed topoi and the naive cotangent complex
    • Section 91.14: Deformations of algebraic spaces
    • Section 91.15: Deformations of complexes
    • Section 91.16: Deformations of complexes on ringed topoi
  • Chapter 92: The Cotangent Complex
    • Section 92.1: Introduction
    • Section 92.2: Advice for the reader
    • Section 92.3: The cotangent complex of a ring map
    • Section 92.4: Simplicial resolutions and derived lower shriek
    • Section 92.5: Constructing a resolution
    • Section 92.6: Functoriality
    • Section 92.7: The fundamental triangle
    • Section 92.8: Localization and ├ętale ring maps
    • Section 92.9: Smooth ring maps
    • Section 92.10: Positive characteristic
    • Section 92.11: Comparison with the naive cotangent complex
    • Section 92.12: A spectral sequence of Quillen
    • Section 92.13: Comparison with Lichtenbaum-Schlessinger
    • Section 92.14: The cotangent complex of a local complete intersection
    • Section 92.15: Tensor products and the cotangent complex
    • Section 92.16: Deformations of ring maps and the cotangent complex
    • Section 92.17: The Atiyah class of a module
    • Section 92.18: The cotangent complex
    • Section 92.19: The Atiyah class of a sheaf of modules
    • Section 92.20: The cotangent complex of a morphism of ringed spaces
    • Section 92.21: Deformations of ringed spaces and the cotangent complex
    • Section 92.22: The cotangent complex of a morphism of ringed topoi
    • Section 92.23: Deformations of ringed topoi and the cotangent complex
    • Section 92.24: The cotangent complex of a morphism of schemes
    • Section 92.25: The cotangent complex of a scheme over a ring
    • Section 92.26: The cotangent complex of a morphism of algebraic spaces
    • Section 92.27: The cotangent complex of an algebraic space over a ring
    • Section 92.28: Fibre products of algebraic spaces and the cotangent complex
  • Chapter 93: Deformation Problems
    • Section 93.1: Introduction
    • Section 93.2: Examples of deformation problems
    • Section 93.3: General outline
    • Section 93.4: Finite projective modules
    • Section 93.5: Representations of a group
    • Section 93.6: Continuous representations
    • Section 93.7: Graded algebras
    • Section 93.8: Rings
    • Section 93.9: Schemes
    • Section 93.10: Morphisms of Schemes
    • Section 93.11: Algebraic spaces
    • Section 93.12: Deformations of completions
    • Section 93.13: Deformations of localizations
    • Section 93.14: Deformations of henselizations
    • Section 93.15: Application to isolated singularities
    • Section 93.16: Unobstructed deformation problems
    • Section 93.17: Smoothings