91 The Cotangent Complex

Section 91.1: Introduction
Section 91.2: Advice for the reader
Section 91.3: The cotangent complex of a ring map
- Definition 91.3.1
- Definition 91.3.2
- Remark 91.3.3
- Lemma 91.3.4
Section 91.4: Simplicial resolutions and derived lower shriek
- Equation 91.4.0.1
- Lemma 91.4.1
- Lemma 91.4.2
- Lemma 91.4.3
- Lemma 91.4.4
- Lemma 91.4.5
- Lemma 91.4.6
- Lemma 91.4.7
Section 91.5: Constructing a resolution
- Lemma 91.5.1
- Proposition 91.5.2
- Lemma 91.5.3
- Lemma 91.5.4
- Remark 91.5.5: Resolutions
- Lemma 91.5.6
- Lemma 91.5.7
- Lemma 91.5.8
- Example 91.5.9
Section 91.6: Functoriality
- Equation 91.6.0.1
- Equation 91.6.0.2
- Lemma 91.6.1
- Lemma 91.6.2
- Remark 91.6.3
- Lemma 91.6.4
Section 91.7: The fundamental triangle
- Equation 91.7.0.1
- Equation 91.7.0.2
- Lemma 91.7.1
- Lemma 91.7.2
- Lemma 91.7.3
- Proposition 91.7.4
- Remark 91.7.5
- Remark 91.7.6
Section 91.8: Localization and étale ring maps
- Lemma 91.8.1
- Lemma 91.8.2
- Lemma 91.8.3
- Lemma 91.8.4
- Lemma 91.8.5
- Lemma 91.8.6
- Lemma 91.8.7
Section 91.9: Smooth ring maps
- Lemma 91.9.1
Section 91.10: Positive characteristic
- Lemma 91.10.1
- Lemma 91.10.2
- Lemma 91.10.3
Section 91.11: Comparison with the naive cotangent complex
- Remark 91.11.1
  - Equation 91.11.1.1
  - Equation 91.11.1.2
- Lemma 91.11.2 slogan
- Lemma 91.11.3
- Remark 91.11.4
- Remark 91.11.5
Section 91.12: A spectral sequence of Quillen
- Lemma 91.12.1
- Remark 91.12.2
- Lemma 91.12.3
- Theorem 91.12.4: Quillen spectral sequence
- Remark 91.12.5
- Remark 91.12.6
Section 91.13: Comparison with Lichtenbaum-Schlessinger
- Equation 91.13.0.1
- Definition 91.13.1
- Lemma 91.13.2
- Lemma 91.13.3
- Remark 91.13.4: Functoriality of the Lichtenbaum-Schlessinger complex
Section 91.14: The cotangent complex of a local complete intersection
- Lemma 91.14.1
- Lemma 91.14.2
- Lemma 91.14.3
- Proposition 91.14.4
Section 91.15: Tensor products and the cotangent complex
- Equation 91.15.0.1
- Lemma 91.15.1
- Lemma 91.15.2
Section 91.16: Deformations of ring maps and the cotangent complex
- Equation 91.16.0.1
- Lemma 91.16.1
Section 91.17: The Atiyah class of a module
- Equation 91.17.0.1
- Definition 91.17.1
Section 91.18: The cotangent complex
- Definition 91.18.1
- Definition 91.18.2
  - Equation 91.18.2.1
- Lemma 91.18.3
- Lemma 91.18.4
- Remark 91.18.5
- Lemma 91.18.6
- Lemma 91.18.7
- Lemma 91.18.8
- Lemma 91.18.9
- Lemma 91.18.10
Section 91.19: The Atiyah class of a sheaf of modules
- Equation 91.19.0.1
- Equation 91.19.0.2
- Definition 91.19.1
Section 91.20: The cotangent complex of a morphism of ringed spaces
- Definition 91.20.1
- Lemma 91.20.2
- Lemma 91.20.3
- Lemma 91.20.4
Section 91.21: Deformations of ringed spaces and the cotangent complex
- Equation 91.21.0.1
- Lemma 91.21.1
Section 91.22: The cotangent complex of a morphism of ringed topoi
- Definition 91.22.1
- Lemma 91.22.2
- Lemma 91.22.3
- Lemma 91.22.4
Section 91.23: Deformations of ringed topoi and the cotangent complex
- Equation 91.23.0.1
- Lemma 91.23.1
Section 91.24: The cotangent complex of a morphism of schemes
- Definition 91.24.1
- Lemma 91.24.2
- Lemma 91.24.3
Section 91.25: The cotangent complex of a scheme over a ring
- Equation 91.25.0.1
- Lemma 91.25.1
  - Equation 91.25.1.1
- Remark 91.25.2
- Lemma 91.25.3
Section 91.26: The cotangent complex of a morphism of algebraic spaces
- Definition 91.26.1
- Lemma 91.26.2
- Lemma 91.26.3
- Lemma 91.26.4
Section 91.27: The cotangent complex of an algebraic space over a ring
- Equation 91.27.0.1
- Lemma 91.27.1
  - Equation 91.27.1.1
- Remark 91.27.2
- Lemma 91.27.3
Section 91.28: Fibre products of algebraic spaces and the cotangent complex
- Equation 91.28.0.1
- Lemma 91.28.1
- Lemma 91.28.2