90 The Cotangent Complex

Section 90.1: Introduction
Section 90.2: Advice for the reader
Section 90.3: The cotangent complex of a ring map
- Definition 90.3.1
- Definition 90.3.2
- Remark 90.3.3
- Lemma 90.3.4
Section 90.4: Simplicial resolutions and derived lower shriek
- Equation 90.4.0.1
- Lemma 90.4.1
- Lemma 90.4.2
- Lemma 90.4.3
- Lemma 90.4.4
- Lemma 90.4.5
- Lemma 90.4.6
- Lemma 90.4.7
Section 90.5: Constructing a resolution
- Lemma 90.5.1
- Proposition 90.5.2
- Lemma 90.5.3
- Lemma 90.5.4
- Remark 90.5.5: Resolutions
- Lemma 90.5.6
- Lemma 90.5.7
- Lemma 90.5.8
- Example 90.5.9
Section 90.6: Functoriality
- Equation 90.6.0.1
- Equation 90.6.0.2
- Lemma 90.6.1
- Lemma 90.6.2
- Remark 90.6.3
- Lemma 90.6.4
Section 90.7: The fundamental triangle
- Equation 90.7.0.1
- Equation 90.7.0.2
- Lemma 90.7.1
- Lemma 90.7.2
- Lemma 90.7.3
- Proposition 90.7.4
- Remark 90.7.5
- Remark 90.7.6
Section 90.8: Localization and étale ring maps
- Lemma 90.8.1
- Lemma 90.8.2
- Lemma 90.8.3
- Lemma 90.8.4
- Lemma 90.8.5
- Lemma 90.8.6
- Lemma 90.8.7
Section 90.9: Smooth ring maps
- Lemma 90.9.1
Section 90.10: Comparison with the naive cotangent complex
- Remark 90.10.1
  - Equation 90.10.1.1
  - Equation 90.10.1.2
- Lemma 90.10.2 slogan
- Lemma 90.10.3
- Remark 90.10.4
- Remark 90.10.5
Section 90.11: A spectral sequence of Quillen
- Lemma 90.11.1
- Remark 90.11.2
- Lemma 90.11.3
- Theorem 90.11.4: Quillen spectral sequence
- Remark 90.11.5
- Remark 90.11.6
Section 90.12: Comparison with Lichtenbaum-Schlessinger
- Equation 90.12.0.1
- Definition 90.12.1
- Lemma 90.12.2
- Lemma 90.12.3
- Remark 90.12.4: Functoriality of the Lichtenbaum-Schlessinger complex
Section 90.13: The cotangent complex of a local complete intersection
- Lemma 90.13.1
- Lemma 90.13.2
- Lemma 90.13.3
- Proposition 90.13.4
Section 90.14: Tensor products and the cotangent complex
- Equation 90.14.0.1
- Lemma 90.14.1
- Lemma 90.14.2
Section 90.15: Deformations of ring maps and the cotangent complex
- Equation 90.15.0.1
- Lemma 90.15.1
Section 90.16: The Atiyah class of a module
- Equation 90.16.0.1
- Definition 90.16.1
Section 90.17: The cotangent complex
- Definition 90.17.1
- Definition 90.17.2
  - Equation 90.17.2.1
- Lemma 90.17.3
- Lemma 90.17.4
- Remark 90.17.5
- Lemma 90.17.6
- Lemma 90.17.7
- Lemma 90.17.8
- Lemma 90.17.9
- Lemma 90.17.10
Section 90.18: The Atiyah class of a sheaf of modules
- Equation 90.18.0.1
- Equation 90.18.0.2
- Definition 90.18.1
Section 90.19: The cotangent complex of a morphism of ringed spaces
- Definition 90.19.1
- Lemma 90.19.2
- Lemma 90.19.3
- Lemma 90.19.4
Section 90.20: Deformations of ringed spaces and the cotangent complex
- Equation 90.20.0.1
- Lemma 90.20.1
Section 90.21: The cotangent complex of a morphism of ringed topoi
- Definition 90.21.1
- Lemma 90.21.2
- Lemma 90.21.3
- Lemma 90.21.4
Section 90.22: Deformations of ringed topoi and the cotangent complex
- Equation 90.22.0.1
- Lemma 90.22.1
Section 90.23: The cotangent complex of a morphism of schemes
- Definition 90.23.1
- Lemma 90.23.2
- Lemma 90.23.3
Section 90.24: The cotangent complex of a scheme over a ring
- Equation 90.24.0.1
- Lemma 90.24.1
  - Equation 90.24.1.1
- Remark 90.24.2
- Lemma 90.24.3
Section 90.25: The cotangent complex of a morphism of algebraic spaces
- Definition 90.25.1
- Lemma 90.25.2
- Lemma 90.25.3
- Lemma 90.25.4
Section 90.26: The cotangent complex of an algebraic space over a ring
- Equation 90.26.0.1
- Lemma 90.26.1
  - Equation 90.26.1.1
- Remark 90.26.2
- Lemma 90.26.3
Section 90.27: Fibre products of algebraic spaces and the cotangent complex
- Equation 90.27.0.1
- Lemma 90.27.1
- Lemma 90.27.2