98 Artin's Axioms

Section 98.1: Introduction
Section 98.2: Conventions
Section 98.3: Predeformation categories
- Equation 98.3.0.1
- Equation 98.3.0.2
- Lemma 98.3.1
  - Equation 98.3.1.1
- Lemma 98.3.2
- Lemma 98.3.3
Section 98.4: Pushouts and stacks
- Lemma 98.4.1 slogan
Section 98.5: The Rim-Schlessinger condition
- Definition 98.5.1
- Lemma 98.5.2
- Lemma 98.5.3
  - Equation 98.5.3.1
Section 98.6: Deformation categories
- Lemma 98.6.1
- Remark 98.6.2
Section 98.7: Change of field
- Lemma 98.7.1
Section 98.8: Tangent spaces
- Equation 98.8.0.1
- Equation 98.8.0.2
- Lemma 98.8.1
- Lemma 98.8.2
Section 98.9: Formal objects
- Definition 98.9.1
- Remark 98.9.2
- Lemma 98.9.3
  - Equation 98.9.3.1
- Definition 98.9.4
- Lemma 98.9.5
- Lemma 98.9.6
Section 98.10: Approximation
- Lemma 98.10.1
  - Equation 98.10.1.1
Section 98.11: Limit preserving
- Definition 98.11.1
- Lemma 98.11.2
- Lemma 98.11.3
- Lemma 98.11.4
Section 98.12: Versality
- Definition 98.12.1
  - Equation 98.12.1.1
- Definition 98.12.2
- Lemma 98.12.3
- Lemma 98.12.4
- Lemma 98.12.5
- Lemma 98.12.6
- Lemma 98.12.7
- Example 98.12.8
Section 98.13: Openness of versality
- Definition 98.13.1
- Example 98.13.2
  - Equation 98.13.2.1
- Lemma 98.13.3
- Lemma 98.13.4
- Lemma 98.13.5
- Lemma 98.13.6
- Lemma 98.13.7
- Lemma 98.13.8
Section 98.14: Axioms
Section 98.15: Axioms for functors
Section 98.16: Algebraic spaces
- Proposition 98.16.1
- Lemma 98.16.2
Section 98.17: Algebraic stacks
- Lemma 98.17.1
- Proposition 98.17.2
Section 98.18: Strong Rim-Schlessinger
- Definition 98.18.1
- Lemma 98.18.2
- Lemma 98.18.3
Section 98.19: Versality and generalizations
- Lemma 98.19.1
- Lemma 98.19.2
Section 98.20: Strong formal effectiveness
- Lemma 98.20.1
- Remark 98.20.2: Strong effectiveness
- Lemma 98.20.3
- Remark 98.20.4
Section 98.21: Infinitesimal deformations
- Lemma 98.21.1
- Lemma 98.21.2
- Remark 98.21.3: Functoriality
- Remark 98.21.4: Automorphisms
- Remark 98.21.5
- Lemma 98.21.6
- Remark 98.21.7: Compatibility with previous tangent spaces
- Remark 98.21.8: Canonical element
- Remark 98.21.9: Canonical automorphism
Section 98.22: Obstruction theories
- Definition 98.22.1
- Lemma 98.22.2 reference
- Example 98.22.3
- Example 98.22.4: Key example
Section 98.23: Naive obstruction theories
- Lemma 98.23.1
- Lemma 98.23.2
- Lemma 98.23.3
- Lemma 98.23.4
- Definition 98.23.5
- Lemma 98.23.6
- Lemma 98.23.7
Section 98.24: A dual notion
- Example 98.24.1
- Situation 98.24.2
- Lemma 98.24.3
- Lemma 98.24.4
Section 98.25: Limit preserving functors on Noetherian schemes
- Remark 98.25.1: Warning
- Lemma 98.25.2
- Lemma 98.25.3
- Lemma 98.25.4
Section 98.26: Algebraic spaces in the Noetherian setting
- Proposition 98.26.1
Section 98.27: Artin's theorem on contractions
- Situation 98.27.1
- Remark 98.27.2
- Remark 98.27.3
- Lemma 98.27.4
- Lemma 98.27.5
- Lemma 98.27.6
- Remark 98.27.7
- Lemma 98.27.8
- Lemma 98.27.9
- Lemma 98.27.10
- Lemma 98.27.11
- Lemma 98.27.12
- Lemma 98.27.13
- Lemma 98.27.14
- Lemma 98.27.15
- Lemma 98.27.16
- Theorem 98.27.17 reference
- Remark 98.27.18