60 Crystalline Cohomology

Section 60.1: Introduction
Section 60.2: Divided power envelope
- Lemma 60.2.1
- Definition 60.2.2
  - Equation 60.2.2.1
- Lemma 60.2.3
- Lemma 60.2.4
- Lemma 60.2.5
- Lemma 60.2.6
- Lemma 60.2.7
  - Equation 60.2.7.1
Section 60.3: Some explicit divided power thickenings
- Lemma 60.3.1
- Lemma 60.3.2
Section 60.4: Compatibility
- Definition 60.4.1
Section 60.5: Affine crystalline site
- Situation 60.5.1
- Definition 60.5.2
  - Equation 60.5.2.1
- Lemma 60.5.3
- Remark 60.5.4
- Lemma 60.5.5
- Lemma 60.5.6
- Lemma 60.5.7
Section 60.6: Module of differentials
- Definition 60.6.1
- Lemma 60.6.2
- Lemma 60.6.3
- Remark 60.6.4
- Lemma 60.6.5
- Lemma 60.6.6
- Remark 60.6.7
- Remark 60.6.8
- Remark 60.6.9
  - Equation 60.6.9.1
- Lemma 60.6.10
Section 60.7: Divided power schemes
- Definition 60.7.1
- Definition 60.7.2
- Definition 60.7.3
- Lemma 60.7.4
- Situation 60.7.5
Section 60.8: The big crystalline site
- Definition 60.8.1
  - Equation 60.8.1.1
  - Equation 60.8.1.2
- Lemma 60.8.2
- Lemma 60.8.3
- Definition 60.8.4
- Remark 60.8.5: Functoriality
- Remark 60.8.6: Comparison with Zariski site
- Remark 60.8.7: Structure morphism
- Remark 60.8.8: Compatibilities
Section 60.9: The crystalline site
- Definition 60.9.1
  - Equation 60.9.1.1
  - Equation 60.9.1.2
- Lemma 60.9.2
- Remark 60.9.3: Functoriality
- Remark 60.9.4: Comparison with Zariski site
- Lemma 60.9.5
- Remark 60.9.6: Structure morphism
- Remark 60.9.7: Compatibilities
Section 60.10: Sheaves on the crystalline site
- Equation 60.10.0.1
- Equation 60.10.0.2
Section 60.11: Crystals in modules
- Definition 60.11.1
- Lemma 60.11.2
- Definition 60.11.3
- Remark 60.11.4
Section 60.12: Sheaf of differentials
- Definition 60.12.1
- Lemma 60.12.2
- Lemma 60.12.3
- Lemma 60.12.4
- Lemma 60.12.5
- Lemma 60.12.6
Section 60.13: Two universal thickenings
- Remark 60.13.1
- Remark 60.13.2
Section 60.14: The de Rham complex
Section 60.15: Connections
- Lemma 60.15.1
Section 60.16: Cosimplicial algebra
- Lemma 60.16.1
Section 60.17: Crystals in quasi-coherent modules
- Equation 60.17.0.1
- Equation 60.17.0.2
- Equation 60.17.0.3
- Equation 60.17.0.4
- Equation 60.17.0.5
- Equation 60.17.0.6
- Lemma 60.17.1
  - Equation 60.17.1.1
- Lemma 60.17.2
- Lemma 60.17.3
- Proposition 60.17.4
- Lemma 60.17.5
- Remark 60.17.6
Section 60.18: General remarks on cohomology
- Lemma 60.18.1
- Lemma 60.18.2
- Lemma 60.18.3
- Lemma 60.18.4
Section 60.19: Cosimplicial preparations
- Example 60.19.1
  - Equation 60.19.1.1
- Lemma 60.19.2
- Lemma 60.19.3
Section 60.20: Divided power Poincaré lemma
- Lemma 60.20.1
- Lemma 60.20.2
Section 60.21: Cohomology in the affine case
- Proposition 60.21.1
- Lemma 60.21.2
- Proposition 60.21.3
- Lemma 60.21.4
Section 60.22: Two counter examples
- Example 60.22.1
- Example 60.22.2
Section 60.23: Applications
- Proposition 60.23.1
- Remark 60.23.2
  - Equation 60.23.2.1
Section 60.24: Some further results
- Remark 60.24.1: Higher direct images
  - Equation 60.24.1.1
- Remark 60.24.2: Mayer-Vietoris
- Remark 60.24.3: Čech complex
- Remark 60.24.4: Alternating Čech complex
- Remark 60.24.5: Quasi-coherence
- Remark 60.24.6: Boundedness
- Remark 60.24.7: Specific boundedness
- Remark 60.24.8: Base change map
  - Equation 60.24.8.1
- Remark 60.24.9: Base change isomorphism
- Remark 60.24.10: Rlim
- Remark 60.24.11: Comparison
- Remark 60.24.12: Perfectness
- Remark 60.24.13: Complete perfectness
- Remark 60.24.14: Complete comparison
Section 60.25: Pulling back along purely inseparable maps
- Lemma 60.25.1
- Example 60.25.2
- Lemma 60.25.3
- Lemma 60.25.4
- Lemma 60.25.5
- Lemma 60.25.6
Section 60.26: Frobenius action on crystalline cohomology
- Situation 60.26.1
- Definition 60.26.2
- Remark 60.26.3
- Theorem 60.26.4
  - Equation 60.26.4.1
  - Equation 60.26.4.2
  - Equation 60.26.4.3