
Chapter 87: Formal Deformation Theory

Section 87.1: Introduction

Section 87.2: Notation and Conventions

Section 87.3: The base category

Section 87.4: The completed base category

Section 87.5: Categories cofibered in groupoids

Section 87.6: Prorepresentable functors and predeformation categories

Section 87.7: Formal objects and completion categories

Section 87.8: Smooth morphisms

Section 87.9: Smooth or unobstructed categories

Section 87.10: Schlessinger's conditions

Section 87.11: Tangent spaces of functors

Section 87.12: Tangent spaces of predeformation categories

Section 87.13: Versal formal objects

Section 87.14: Minimal versal formal objects

Section 87.15: Miniversal formal objects and tangent spaces

Section 87.16: RimSchlessinger conditions and deformation categories

Section 87.17: Lifts of objects

Section 87.18: Schlessinger's theorem on prorepresentable functors

Section 87.19: Infinitesimal automorphisms

Section 87.20: Applications

Section 87.21: Groupoids in functors on an arbitrary category

Section 87.22: Groupoids in functors on the base category

Section 87.23: Smooth groupoids in functors on the base category

Section 87.24: Deformation categories as quotients of groupoids in functors

Section 87.25: Presentations of categories cofibered in groupoids

Section 87.26: Presentations of deformation categories

Section 87.27: Remarks regarding minimality

Section 87.28: Uniqueness of versal rings

Section 87.29: Change of residue field

Chapter 88: Deformation Theory

Section 88.1: Introduction

Section 88.2: Deformations of rings and the naive cotangent complex

Section 88.3: Thickenings of ringed spaces

Section 88.4: Modules on first order thickenings of ringed spaces

Section 88.5: Infinitesimal deformations of modules on ringed spaces

Section 88.6: Application to flat modules on flat thickenings of ringed spaces

Section 88.7: Deformations of ringed spaces and the naive cotangent complex

Section 88.8: Deformations of schemes

Section 88.9: Thickenings of ringed topoi

Section 88.10: Modules on first order thickenings of ringed topoi

Section 88.11: Infinitesimal deformations of modules on ringed topoi

Section 88.12: Application to flat modules on flat thickenings of ringed topoi

Section 88.13: Deformations of ringed topoi and the naive cotangent complex

Section 88.14: Deformations of algebraic spaces

Section 88.15: Deformations of complexes

Section 88.16: Deformations of complexes on ringed topoi

Chapter 89: The Cotangent Complex

Section 89.1: Introduction

Section 89.2: Advice for the reader

Section 89.3: The cotangent complex of a ring map

Section 89.4: Simplicial resolutions and derived lower shriek

Section 89.5: Constructing a resolution

Section 89.6: Functoriality

Section 89.7: The fundamental triangle

Section 89.8: Localization and étale ring maps

Section 89.9: Smooth ring maps

Section 89.10: Comparison with the naive cotangent complex

Section 89.11: A spectral sequence of Quillen

Section 89.12: Comparison with LichtenbaumSchlessinger

Section 89.13: The cotangent complex of a local complete intersection

Section 89.14: Tensor products and the cotangent complex

Section 89.15: Deformations of ring maps and the cotangent complex

Section 89.16: The Atiyah class of a module

Section 89.17: The cotangent complex

Section 89.18: The Atiyah class of a sheaf of modules

Section 89.19: The cotangent complex of a morphism of ringed spaces

Section 89.20: Deformations of ringed spaces and the cotangent complex

Section 89.21: The cotangent complex of a morphism of ringed topoi

Section 89.22: Deformations of ringed topoi and the cotangent complex

Section 89.23: The cotangent complex of a morphism of schemes

Section 89.24: The cotangent complex of a scheme over a ring

Section 89.25: The cotangent complex of a morphism of algebraic spaces

Section 89.26: The cotangent complex of an algebraic space over a ring

Section 89.27: Fibre products of algebraic spaces and the cotangent complex

Chapter 90: Deformation Problems

Section 90.1: Introduction

Section 90.2: Examples of deformation problems

Section 90.3: General outline

Section 90.4: Finite projective modules

Section 90.5: Representations of a group

Section 90.6: Continuous representations

Section 90.7: Graded algebras

Section 90.8: Rings

Section 90.9: Schemes

Section 90.10: Morphisms of Schemes

Section 90.11: Algebraic spaces

Section 90.12: Deformations of completions

Section 90.13: Deformations of localizations

Section 90.14: Deformations of henselizations

Section 90.15: Application to isolated singularities

Section 90.16: Unobstructed deformation problems

Section 90.17: Smoothings