The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

6 Deformation Theory

  • Chapter 82: Formal Deformation Theory
    • Section 82.1: Introduction
    • Section 82.2: Notation and Conventions
    • Section 82.3: The base category
    • Section 82.4: The completed base category
    • Section 82.5: Categories cofibered in groupoids
    • Section 82.6: Prorepresentable functors and predeformation categories
    • Section 82.7: Formal objects and completion categories
    • Section 82.8: Smooth morphisms
    • Section 82.9: Smooth or unobstructed categories
    • Section 82.10: Schlessinger's conditions
    • Section 82.11: Tangent spaces of functors
    • Section 82.12: Tangent spaces of predeformation categories
    • Section 82.13: Versal formal objects
    • Section 82.14: Minimal versal formal objects
    • Section 82.15: Miniversal formal objects and tangent spaces
    • Section 82.16: Rim-Schlessinger conditions and deformation categories
    • Section 82.17: Lifts of objects
    • Section 82.18: Schlessinger's theorem on prorepresentable functors
    • Section 82.19: Infinitesimal automorphisms
    • Section 82.20: Applications
    • Section 82.21: Groupoids in functors on an arbitrary category
    • Section 82.22: Groupoids in functors on the base category
    • Section 82.23: Smooth groupoids in functors on the base category
    • Section 82.24: Deformation categories as quotients of groupoids in functors
    • Section 82.25: Presentations of categories cofibered in groupoids
    • Section 82.26: Presentations of deformation categories
    • Section 82.27: Remarks regarding minimality
    • Section 82.28: Uniqueness of versal rings
    • Section 82.29: Change of residue field
  • Chapter 83: Deformation Theory
    • Section 83.1: Introduction
    • Section 83.2: Deformations of rings and the naive cotangent complex
    • Section 83.3: Thickenings of ringed spaces
    • Section 83.4: Modules on first order thickenings of ringed spaces
    • Section 83.5: Infinitesimal deformations of modules on ringed spaces
    • Section 83.6: Application to flat modules on flat thickenings of ringed spaces
    • Section 83.7: Deformations of ringed spaces and the naive cotangent complex
    • Section 83.8: Deformations of schemes
    • Section 83.9: Thickenings of ringed topoi
    • Section 83.10: Modules on first order thickenings of ringed topoi
    • Section 83.11: Infinitesimal deformations of modules on ringed topoi
    • Section 83.12: Application to flat modules on flat thickenings of ringed topoi
    • Section 83.13: Deformations of ringed topoi and the naive cotangent complex
    • Section 83.14: Deformations of algebraic spaces
    • Section 83.15: Deformations of complexes
    • Section 83.16: Deformations of complexes on ringed topoi
  • Chapter 84: The Cotangent Complex
    • Section 84.1: Introduction
    • Section 84.2: Advice for the reader
    • Section 84.3: The cotangent complex of a ring map
    • Section 84.4: Simplicial resolutions and derived lower shriek
    • Section 84.5: Constructing a resolution
    • Section 84.6: Functoriality
    • Section 84.7: The fundamental triangle
    • Section 84.8: Localization and ├ętale ring maps
    • Section 84.9: Smooth ring maps
    • Section 84.10: Comparison with the naive cotangent complex
    • Section 84.11: A spectral sequence of Quillen
    • Section 84.12: Comparison with Lichtenbaum-Schlessinger
    • Section 84.13: The cotangent complex of a local complete intersection
    • Section 84.14: Tensor products and the cotangent complex
    • Section 84.15: Deformations of ring maps and the cotangent complex
    • Section 84.16: The Atiyah class of a module
    • Section 84.17: The cotangent complex
    • Section 84.18: The Atiyah class of a sheaf of modules
    • Section 84.19: The cotangent complex of a morphism of ringed spaces
    • Section 84.20: Deformations of ringed spaces and the cotangent complex
    • Section 84.21: The cotangent complex of a morphism of ringed topoi
    • Section 84.22: Deformations of ringed topoi and the cotangent complex
    • Section 84.23: The cotangent complex of a morphism of schemes
    • Section 84.24: The cotangent complex of a scheme over a ring
    • Section 84.25: The cotangent complex of a morphism of algebraic spaces
    • Section 84.26: The cotangent complex of an algebraic space over a ring
    • Section 84.27: Fibre products of algebraic spaces and the cotangent complex
  • Chapter 85: Deformation Problems
    • Section 85.1: Introduction
    • Section 85.2: Examples of deformation problems
    • Section 85.3: General outline
    • Section 85.4: Finite projective modules
    • Section 85.5: Representations of a group
    • Section 85.6: Continuous representations
    • Section 85.7: Graded algebras
    • Section 85.8: Rings
    • Section 85.9: Schemes
    • Section 85.10: Morphisms of Schemes
    • Section 85.11: Algebraic spaces
    • Section 85.12: Deformations of completions
    • Section 85.13: Deformations of localizations
    • Section 85.14: Deformations of henselizations
    • Section 85.15: Application to isolated singularities
    • Section 85.16: Unobstructed deformation problems
    • Section 85.17: Smoothings