88 Formal Deformation Theory

Section 88.1: Introduction
Section 88.2: Notation and Conventions
Section 88.3: The base category
- Definition 88.3.1
- Definition 88.3.2
- Lemma 88.3.3
  - Equation 88.3.3.1
- Lemma 88.3.4
- Lemma 88.3.5
- Definition 88.3.6
- Example 88.3.7
- Lemma 88.3.8
- Definition 88.3.9
- Lemma 88.3.10
  - Equation 88.3.10.1
  - Equation 88.3.10.2
  - Equation 88.3.10.3
- Lemma 88.3.11
- Lemma 88.3.12
- Example 88.3.13
Section 88.4: The completed base category
- Definition 88.4.1
- Lemma 88.4.2
- Lemma 88.4.3
- Lemma 88.4.4
- Lemma 88.4.5
- Lemma 88.4.6
- Lemma 88.4.7
- Lemma 88.4.8
- Lemma 88.4.9
Section 88.5: Categories cofibered in groupoids
- Definition 88.5.1
- Remarks 88.5.2
Section 88.6: Prorepresentable functors and predeformation categories
- Definition 88.6.1
- Definition 88.6.2
- Remark 88.6.3
- Remark 88.6.4
Section 88.7: Formal objects and completion categories
- Definition 88.7.1
- Lemma 88.7.2
- Definition 88.7.3
- Lemma 88.7.4
- Remark 88.7.5
- Remark 88.7.6
- Remark 88.7.7
- Remark 88.7.8
- Remark 88.7.9
- Remark 88.7.10
- Remark 88.7.11
- Remark 88.7.12
  - Equation 88.7.12.1
- Remark 88.7.13
- Remark 88.7.14
Section 88.8: Smooth morphisms
- Definition 88.8.1
- Lemma 88.8.2
- Remark 88.8.3
- Remark 88.8.4
- Remark 88.8.5
- Lemma 88.8.6
- Lemma 88.8.7
- Lemma 88.8.8
- Definition 88.8.9
- Remark 88.8.10
- Lemma 88.8.11
Section 88.9: Smooth or unobstructed categories
- Definition 88.9.1
- Remark 88.9.2
- Lemma 88.9.3
- Lemma 88.9.4
- Lemma 88.9.5
- Example 88.9.6
Section 88.10: Schlessinger's conditions
- Definition 88.10.1
  - Equation 88.10.1.1
- Lemma 88.10.2
- Remark 88.10.3
- Lemma 88.10.4
- Lemma 88.10.5
- Lemma 88.10.6
- Lemma 88.10.7
- Lemma 88.10.8
Section 88.11: Tangent spaces of functors
- Definition 88.11.1
- Remark 88.11.2
- Remark 88.11.3
- Lemma 88.11.4
- Lemma 88.11.5
- Lemma 88.11.6
- Lemma 88.11.7
- Lemma 88.11.8
- Definition 88.11.9
- Example 88.11.10
- Example 88.11.11
- Example 88.11.12
- Lemma 88.11.13
- Example 88.11.14
- Lemma 88.11.15
Section 88.12: Tangent spaces of predeformation categories
- Definition 88.12.1
- Lemma 88.12.2
- Definition 88.12.3
- Lemma 88.12.4
- Remark 88.12.5
- Lemma 88.12.6
  - Equation 88.12.6.1
  - Equation 88.12.6.2
Section 88.13: Versal formal objects
- Lemma 88.13.1
- Lemma 88.13.2
- Lemma 88.13.3
- Lemma 88.13.4
- Remark 88.13.5
Section 88.14: Minimal versal formal objects
- Lemma 88.14.1
- Lemma 88.14.2
- Lemma 88.14.3
- Definition 88.14.4
- Lemma 88.14.5
Section 88.15: Miniversal formal objects and tangent spaces
- Equation 88.15.0.1
- Equation 88.15.0.2
- Lemma 88.15.1
- Lemma 88.15.2
- Example 88.15.3
- Lemma 88.15.4
- Theorem 88.15.5
- Remark 88.15.6
- Remark 88.15.7
- Example 88.15.8
Section 88.16: Rim-Schlessinger conditions and deformation categories
- Definition 88.16.1
- Lemma 88.16.2
- Lemma 88.16.3
- Lemma 88.16.4
- Remark 88.16.5
- Lemma 88.16.6
- Lemma 88.16.7
- Definition 88.16.8
- Remark 88.16.9
- Example 88.16.10
- Lemma 88.16.11
- Lemma 88.16.12
Section 88.17: Lifts of objects
- Definition 88.17.1
- Remark 88.17.2
- Remark 88.17.3
- Remark 88.17.4
- Lemma 88.17.5
  - Equation 88.17.5.1
  - Equation 88.17.5.2
  - Equation 88.17.5.3
- Remark 88.17.6
Section 88.18: Schlessinger's theorem on prorepresentable functors
- Lemma 88.18.1
- Theorem 88.18.2
Section 88.19: Infinitesimal automorphisms
- Definition 88.19.1
- Definition 88.19.2
- Remark 88.19.3
- Remark 88.19.4
- Definition 88.19.5
- Lemma 88.19.6
- Lemma 88.19.7
- Remark 88.19.8
- Lemma 88.19.9
- Lemma 88.19.10
- Lemma 88.19.11
- Lemma 88.19.12
- Lemma 88.19.13
Section 88.20: Applications
- Lemma 88.20.1
- Lemma 88.20.2
- Lemma 88.20.3
- Lemma 88.20.4
Section 88.21: Groupoids in functors on an arbitrary category
- Definition 88.21.1
- Remark 88.21.2
- Remark 88.21.3
- Definition 88.21.4
- Remark 88.21.5
- Remark 88.21.6
- Definition 88.21.7
- Remark 88.21.8
- Definition 88.21.9
Section 88.22: Groupoids in functors on the base category
- Definition 88.22.1
- Definition 88.22.2
- Remark 88.22.3
- Lemma 88.22.4
Section 88.23: Smooth groupoids in functors on the base category
- Definition 88.23.1
- Remark 88.23.2
- Remark 88.23.3
- Lemma 88.23.4
Section 88.24: Deformation categories as quotients of groupoids in functors
- Lemma 88.24.1
- Lemma 88.24.2
Section 88.25: Presentations of categories cofibered in groupoids
- Definition 88.25.1
- Lemma 88.25.2
- Lemma 88.25.3
Section 88.26: Presentations of deformation categories
- Lemma 88.26.1
- Lemma 88.26.2
- Lemma 88.26.3
- Theorem 88.26.4
Section 88.27: Remarks regarding minimality
- Definition 88.27.1
- Lemma 88.27.2
- Lemma 88.27.3
- Lemma 88.27.4
- Lemma 88.27.5
- Lemma 88.27.6
- Lemma 88.27.7
- Theorem 88.27.8
Section 88.28: Uniqueness of versal rings
- Lemma 88.28.1
- Lemma 88.28.2
- Lemma 88.28.3
- Lemma 88.28.4
Section 88.29: Change of residue field
- Situation 88.29.1
  - Equation 88.29.1.1
- Lemma 88.29.2
- Lemma 88.29.3
- Lemma 88.29.4
- Lemma 88.29.5
- Lemma 88.29.6