
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: RimSchlessinger 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 étale 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 LichtenbaumSchlessinger

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