
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: RimSchlessinger 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 étale 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 LichtenbaumSchlessinger

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