87 Formal Deformation Theory

Section 87.1: Introduction
Section 87.2: Notation and Conventions
Section 87.3: The base category
- Definition 87.3.1
- Definition 87.3.2
- Lemma 87.3.3
  - Equation 87.3.3.1
- Lemma 87.3.4
- Lemma 87.3.5
- Definition 87.3.6
- Example 87.3.7
- Lemma 87.3.8
- Definition 87.3.9
- Lemma 87.3.10
  - Equation 87.3.10.1
  - Equation 87.3.10.2
  - Equation 87.3.10.3
- Lemma 87.3.11
- Lemma 87.3.12
- Example 87.3.13
Section 87.4: The completed base category
- Definition 87.4.1
- Lemma 87.4.2
- Lemma 87.4.3
- Lemma 87.4.4
- Lemma 87.4.5
- Lemma 87.4.6
- Lemma 87.4.7
- Lemma 87.4.8
- Lemma 87.4.9
Section 87.5: Categories cofibered in groupoids
- Definition 87.5.1
- Remarks 87.5.2
Section 87.6: Prorepresentable functors and predeformation categories
- Definition 87.6.1
- Definition 87.6.2
- Remark 87.6.3
- Remark 87.6.4
Section 87.7: Formal objects and completion categories
- Definition 87.7.1
- Lemma 87.7.2
- Definition 87.7.3
- Lemma 87.7.4
- Remark 87.7.5
- Remark 87.7.6
- Remark 87.7.7
- Remark 87.7.8
- Remark 87.7.9
- Remark 87.7.10
- Remark 87.7.11
- Remark 87.7.12
  - Equation 87.7.12.1
- Remark 87.7.13
- Remark 87.7.14
Section 87.8: Smooth morphisms
- Definition 87.8.1
- Lemma 87.8.2
- Remark 87.8.3
- Remark 87.8.4
- Remark 87.8.5
- Lemma 87.8.6
- Lemma 87.8.7
- Lemma 87.8.8
- Definition 87.8.9
- Remark 87.8.10
- Lemma 87.8.11
Section 87.9: Smooth or unobstructed categories
- Definition 87.9.1
- Remark 87.9.2
- Lemma 87.9.3
- Lemma 87.9.4
- Lemma 87.9.5
- Example 87.9.6
Section 87.10: Schlessinger's conditions
- Definition 87.10.1
  - Equation 87.10.1.1
- Lemma 87.10.2
- Remark 87.10.3
- Lemma 87.10.4
- Lemma 87.10.5
- Lemma 87.10.6
- Lemma 87.10.7
- Lemma 87.10.8
Section 87.11: Tangent spaces of functors
- Definition 87.11.1
- Remark 87.11.2
- Remark 87.11.3
- Lemma 87.11.4
- Lemma 87.11.5
- Lemma 87.11.6
- Lemma 87.11.7
- Lemma 87.11.8
- Definition 87.11.9
- Example 87.11.10
- Example 87.11.11
- Example 87.11.12
- Lemma 87.11.13
- Example 87.11.14
- Lemma 87.11.15
Section 87.12: Tangent spaces of predeformation categories
- Definition 87.12.1
- Lemma 87.12.2
- Definition 87.12.3
- Lemma 87.12.4
- Remark 87.12.5
- Lemma 87.12.6
  - Equation 87.12.6.1
  - Equation 87.12.6.2
Section 87.13: Versal formal objects
- Lemma 87.13.1
- Lemma 87.13.2
- Lemma 87.13.3
- Lemma 87.13.4
- Remark 87.13.5
Section 87.14: Minimal versal formal objects
- Lemma 87.14.1
- Lemma 87.14.2
- Lemma 87.14.3
- Definition 87.14.4
- Lemma 87.14.5
Section 87.15: Miniversal formal objects and tangent spaces
- Equation 87.15.0.1
- Equation 87.15.0.2
- Lemma 87.15.1
- Lemma 87.15.2
- Example 87.15.3
- Lemma 87.15.4
- Theorem 87.15.5
- Remark 87.15.6
- Remark 87.15.7
- Example 87.15.8
Section 87.16: Rim-Schlessinger conditions and deformation categories
- Definition 87.16.1
- Lemma 87.16.2
- Lemma 87.16.3
- Lemma 87.16.4
- Remark 87.16.5
- Lemma 87.16.6
- Lemma 87.16.7
- Definition 87.16.8
- Remark 87.16.9
- Example 87.16.10
- Lemma 87.16.11
- Lemma 87.16.12
Section 87.17: Lifts of objects
- Definition 87.17.1
- Remark 87.17.2
- Remark 87.17.3
- Remark 87.17.4
- Lemma 87.17.5
  - Equation 87.17.5.1
  - Equation 87.17.5.2
  - Equation 87.17.5.3
- Remark 87.17.6
Section 87.18: Schlessinger's theorem on prorepresentable functors
- Lemma 87.18.1
- Theorem 87.18.2
Section 87.19: Infinitesimal automorphisms
- Definition 87.19.1
- Definition 87.19.2
- Remark 87.19.3
- Remark 87.19.4
- Definition 87.19.5
- Lemma 87.19.6
- Lemma 87.19.7
- Remark 87.19.8
- Lemma 87.19.9
- Lemma 87.19.10
- Lemma 87.19.11
- Lemma 87.19.12
- Lemma 87.19.13
Section 87.20: Applications
- Lemma 87.20.1
- Lemma 87.20.2
- Lemma 87.20.3
- Lemma 87.20.4
Section 87.21: Groupoids in functors on an arbitrary category
- Definition 87.21.1
- Remark 87.21.2
- Remark 87.21.3
- Definition 87.21.4
- Remark 87.21.5
- Remark 87.21.6
- Definition 87.21.7
- Remark 87.21.8
- Definition 87.21.9
Section 87.22: Groupoids in functors on the base category
- Definition 87.22.1
- Definition 87.22.2
- Remark 87.22.3
- Lemma 87.22.4
Section 87.23: Smooth groupoids in functors on the base category
- Definition 87.23.1
- Remark 87.23.2
- Remark 87.23.3
- Lemma 87.23.4
Section 87.24: Deformation categories as quotients of groupoids in functors
- Lemma 87.24.1
- Lemma 87.24.2
Section 87.25: Presentations of categories cofibered in groupoids
- Definition 87.25.1
- Lemma 87.25.2
- Lemma 87.25.3
Section 87.26: Presentations of deformation categories
- Lemma 87.26.1
- Lemma 87.26.2
- Lemma 87.26.3
- Theorem 87.26.4
Section 87.27: Remarks regarding minimality
- Definition 87.27.1
- Lemma 87.27.2
- Lemma 87.27.3
- Lemma 87.27.4
- Lemma 87.27.5
- Lemma 87.27.6
- Lemma 87.27.7
- Theorem 87.27.8
Section 87.28: Uniqueness of versal rings
- Lemma 87.28.1
- Lemma 87.28.2
- Lemma 87.28.3
- Lemma 87.28.4
Section 87.29: Change of residue field
- Situation 87.29.1
  - Equation 87.29.1.1
- Lemma 87.29.2
- Lemma 87.29.3
- Lemma 87.29.4
- Lemma 87.29.5
- Lemma 87.29.6