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
- Definition 89.3.1
- Definition 89.3.2
- Remark 89.3.3
- Lemma 89.3.4
Section 89.4: Simplicial resolutions and derived lower shriek
- Equation 89.4.0.1
- Lemma 89.4.1
- Lemma 89.4.2
- Lemma 89.4.3
- Lemma 89.4.4
- Lemma 89.4.5
- Lemma 89.4.6
- Lemma 89.4.7
Section 89.5: Constructing a resolution
- Lemma 89.5.1
- Proposition 89.5.2
- Lemma 89.5.3
- Lemma 89.5.4
- Remark 89.5.5: Resolutions
- Lemma 89.5.6
- Lemma 89.5.7
- Lemma 89.5.8
- Example 89.5.9
Section 89.6: Functoriality
- Equation 89.6.0.1
- Equation 89.6.0.2
- Lemma 89.6.1
- Lemma 89.6.2
- Remark 89.6.3
- Lemma 89.6.4
Section 89.7: The fundamental triangle
- Equation 89.7.0.1
- Equation 89.7.0.2
- Lemma 89.7.1
- Lemma 89.7.2
- Lemma 89.7.3
- Proposition 89.7.4
- Remark 89.7.5
- Remark 89.7.6
Section 89.8: Localization and étale ring maps
- Lemma 89.8.1
- Lemma 89.8.2
- Lemma 89.8.3
- Lemma 89.8.4
- Lemma 89.8.5
- Lemma 89.8.6
- Lemma 89.8.7
Section 89.9: Smooth ring maps
- Lemma 89.9.1
Section 89.10: Comparison with the naive cotangent complex
- Remark 89.10.1
  - Equation 89.10.1.1
  - Equation 89.10.1.2
- Lemma 89.10.2 slogan
- Lemma 89.10.3
- Remark 89.10.4
- Remark 89.10.5
Section 89.11: A spectral sequence of Quillen
- Lemma 89.11.1
- Remark 89.11.2
- Lemma 89.11.3
- Theorem 89.11.4: Quillen spectral sequence
- Remark 89.11.5
- Remark 89.11.6
Section 89.12: Comparison with Lichtenbaum-Schlessinger
- Equation 89.12.0.1
- Definition 89.12.1
- Lemma 89.12.2
- Lemma 89.12.3
- Remark 89.12.4: Functoriality of the Lichtenbaum-Schlessinger complex
Section 89.13: The cotangent complex of a local complete intersection
- Lemma 89.13.1
- Lemma 89.13.2
- Lemma 89.13.3
- Proposition 89.13.4
Section 89.14: Tensor products and the cotangent complex
- Equation 89.14.0.1
- Lemma 89.14.1
- Lemma 89.14.2
Section 89.15: Deformations of ring maps and the cotangent complex
- Equation 89.15.0.1
- Lemma 89.15.1
Section 89.16: The Atiyah class of a module
- Equation 89.16.0.1
- Definition 89.16.1
Section 89.17: The cotangent complex
- Definition 89.17.1
- Definition 89.17.2
  - Equation 89.17.2.1
- Lemma 89.17.3
- Lemma 89.17.4
- Remark 89.17.5
- Lemma 89.17.6
- Lemma 89.17.7
- Lemma 89.17.8
- Lemma 89.17.9
- Lemma 89.17.10
Section 89.18: The Atiyah class of a sheaf of modules
- Equation 89.18.0.1
- Equation 89.18.0.2
- Definition 89.18.1
Section 89.19: The cotangent complex of a morphism of ringed spaces
- Definition 89.19.1
- Lemma 89.19.2
- Lemma 89.19.3
- Lemma 89.19.4
Section 89.20: Deformations of ringed spaces and the cotangent complex
- Equation 89.20.0.1
- Lemma 89.20.1
Section 89.21: The cotangent complex of a morphism of ringed topoi
- Definition 89.21.1
- Lemma 89.21.2
- Lemma 89.21.3
- Lemma 89.21.4
Section 89.22: Deformations of ringed topoi and the cotangent complex
- Equation 89.22.0.1
- Lemma 89.22.1
Section 89.23: The cotangent complex of a morphism of schemes
- Definition 89.23.1
- Lemma 89.23.2
- Lemma 89.23.3
Section 89.24: The cotangent complex of a scheme over a ring
- Equation 89.24.0.1
- Lemma 89.24.1
  - Equation 89.24.1.1
- Remark 89.24.2
- Lemma 89.24.3
Section 89.25: The cotangent complex of a morphism of algebraic spaces
- Definition 89.25.1
- Lemma 89.25.2
- Lemma 89.25.3
- Lemma 89.25.4
Section 89.26: The cotangent complex of an algebraic space over a ring
- Equation 89.26.0.1
- Lemma 89.26.1
  - Equation 89.26.1.1
- Remark 89.26.2
- Lemma 89.26.3
Section 89.27: Fibre products of algebraic spaces and the cotangent complex
- Equation 89.27.0.1
- Lemma 89.27.1
- Lemma 89.27.2