92 The Cotangent Complex

Section 92.1: Introduction
Section 92.2: Advice for the reader
Section 92.3: The cotangent complex of a ring map
- Definition 92.3.1
- Definition 92.3.2
- Remark 92.3.3
- Lemma 92.3.4
Section 92.4: Simplicial resolutions and derived lower shriek
- Equation 92.4.0.1
- Lemma 92.4.1
- Lemma 92.4.2
- Lemma 92.4.3
- Lemma 92.4.4
- Lemma 92.4.5
- Lemma 92.4.6
- Lemma 92.4.7
Section 92.5: Constructing a resolution
- Lemma 92.5.1
- Proposition 92.5.2
- Lemma 92.5.3
- Lemma 92.5.4
- Remark 92.5.5: Resolutions
- Lemma 92.5.6
- Lemma 92.5.7
- Lemma 92.5.8
- Example 92.5.9
Section 92.6: Functoriality
- Equation 92.6.0.1
- Equation 92.6.0.2
- Lemma 92.6.1
- Lemma 92.6.2
- Remark 92.6.3
- Lemma 92.6.4
Section 92.7: The fundamental triangle
- Equation 92.7.0.1
- Equation 92.7.0.2
- Lemma 92.7.1
- Lemma 92.7.2
- Lemma 92.7.3
- Proposition 92.7.4
- Remark 92.7.5
- Remark 92.7.6
Section 92.8: Localization and étale ring maps
- Lemma 92.8.1
- Lemma 92.8.2
- Lemma 92.8.3
- Lemma 92.8.4
- Lemma 92.8.5
- Lemma 92.8.6
- Lemma 92.8.7
Section 92.9: Smooth ring maps
- Lemma 92.9.1
Section 92.10: Positive characteristic
- Lemma 92.10.1
- Lemma 92.10.2
- Lemma 92.10.3
Section 92.11: Comparison with the naive cotangent complex
- Remark 92.11.1
  - Equation 92.11.1.1
  - Equation 92.11.1.2
- Lemma 92.11.2 slogan
- Lemma 92.11.3
- Remark 92.11.4
- Remark 92.11.5
Section 92.12: A spectral sequence of Quillen
- Lemma 92.12.1
- Remark 92.12.2
- Lemma 92.12.3
- Theorem 92.12.4: Quillen spectral sequence
- Remark 92.12.5
- Remark 92.12.6
Section 92.13: Comparison with Lichtenbaum-Schlessinger
- Equation 92.13.0.1
- Definition 92.13.1
- Lemma 92.13.2
- Lemma 92.13.3
- Remark 92.13.4: Functoriality of the Lichtenbaum-Schlessinger complex
Section 92.14: The cotangent complex of a local complete intersection
- Lemma 92.14.1
- Lemma 92.14.2
- Lemma 92.14.3
- Proposition 92.14.4
Section 92.15: Tensor products and the cotangent complex
- Equation 92.15.0.1
- Lemma 92.15.1
- Lemma 92.15.2
Section 92.16: Deformations of ring maps and the cotangent complex
- Equation 92.16.0.1
- Lemma 92.16.1
Section 92.17: The Atiyah class of a module
- Equation 92.17.0.1
- Definition 92.17.1
Section 92.18: The cotangent complex
- Definition 92.18.1
- Definition 92.18.2
  - Equation 92.18.2.1
- Lemma 92.18.3
- Lemma 92.18.4
- Remark 92.18.5
- Lemma 92.18.6
- Lemma 92.18.7
- Lemma 92.18.8
- Lemma 92.18.9
- Lemma 92.18.10
Section 92.19: The Atiyah class of a sheaf of modules
- Equation 92.19.0.1
- Equation 92.19.0.2
- Definition 92.19.1
Section 92.20: The cotangent complex of a morphism of ringed spaces
- Definition 92.20.1
- Lemma 92.20.2
- Lemma 92.20.3
- Lemma 92.20.4
Section 92.21: Deformations of ringed spaces and the cotangent complex
- Equation 92.21.0.1
- Lemma 92.21.1
Section 92.22: The cotangent complex of a morphism of ringed topoi
- Definition 92.22.1
- Lemma 92.22.2
- Lemma 92.22.3
- Lemma 92.22.4
Section 92.23: Deformations of ringed topoi and the cotangent complex
- Equation 92.23.0.1
- Lemma 92.23.1
Section 92.24: The cotangent complex of a morphism of schemes
- Definition 92.24.1
- Lemma 92.24.2
- Lemma 92.24.3
Section 92.25: The cotangent complex of a scheme over a ring
- Equation 92.25.0.1
- Lemma 92.25.1
  - Equation 92.25.1.1
- Remark 92.25.2
- Lemma 92.25.3
Section 92.26: The cotangent complex of a morphism of algebraic spaces
- Definition 92.26.1
- Lemma 92.26.2
- Lemma 92.26.3
- Lemma 92.26.4
Section 92.27: The cotangent complex of an algebraic space over a ring
- Equation 92.27.0.1
- Lemma 92.27.1
  - Equation 92.27.1.1
- Remark 92.27.2
- Lemma 92.27.3
Section 92.28: Fibre products of algebraic spaces and the cotangent complex
- Equation 92.28.0.1
- Lemma 92.28.1
- Lemma 92.28.2