59 Étale Cohomology

• Section 59.1: Introduction
• Section 59.2: Which sections to skip on a first reading?
• Section 59.3: Prologue
• Section 59.4: The étale topology
  • Definition 59.4.1
  • Remark 59.4.2
• Section 59.5: Feats of the étale topology
• Section 59.6: A computation
• Section 59.7: Nontorsion coefficients
• Section 59.8: Sheaf theory
• Section 59.9: Presheaves
  • Definition 59.9.1
  • Example 59.9.2
  • Lemma 59.9.3: Yoneda slogan
• Section 59.10: Sites
  • Definition 59.10.1
  • Definition 59.10.2
  • Example 59.10.3
• Section 59.11: Sheaves
  • Definition 59.11.1
    • Equation 59.11.1.1
  • Remark 59.11.2
  • Example 59.11.3
  • Definition 59.11.4
• Section 59.12: The example of G-sets
• Section 59.13: Sheafification
  • Definition 59.13.1
  • Theorem 59.13.2
• Section 59.14: Cohomology
  • Theorem 59.14.1
• Section 59.15: The fpqc topology
  • Definition 59.15.1
  • Remark 59.15.2
  • Example 59.15.3
  • Lemma 59.15.4
  • Definition 59.15.5
  • Lemma 59.15.6
  • Example 59.15.7
  • Lemma 59.15.8
  • Remark 59.15.9
  • Example 59.15.10
• Section 59.16: Faithfully flat descent
  • Definition 59.16.1
  • Theorem 59.16.2
  • Lemma 59.16.3
  • Lemma 59.16.4
  • Definition 59.16.5
  • Definition 59.16.6
  • Theorem 59.16.7
  • Remarks 59.16.8
• Section 59.17: Quasi-coherent sheaves
  • Proposition 59.17.1
  • Definition 59.17.2
  • Remark 59.17.3
  • Theorem 59.17.4: Meta theorem on quasi-coherent sheaves
• Section 59.18: Čech cohomology
  • Definition 59.18.1
  • Lemma 59.18.2
  • Lemma 59.18.3: Yoneda Lemma
  • Definition 59.18.4
  • Lemma 59.18.5
  • Lemma 59.18.6
  • Lemma 59.18.7
  • Theorem 59.18.8
  • Remark 59.18.9
• Section 59.19: The Čech-to-cohomology spectral sequence
  • Lemma 59.19.1
  • Theorem 59.19.2
  • Remark 59.19.3
• Section 59.20: Big and small sites of schemes
  • Definition 59.20.1
  • Definition 59.20.2
  • Lemma 59.20.3
  • Definition 59.20.4
  • Lemma 59.20.5
  • Lemma 59.20.6
• Section 59.21: The étale topos
  • Definition 59.21.1
  • Lemma 59.21.2
• Section 59.22: Cohomology of quasi-coherent sheaves
  • Lemma 59.22.1
    • Equation 59.22.1.1
  • Remark 59.22.2
  • Lemma 59.22.3: Locality of cohomology
  • Theorem 59.22.4
  • Remark 59.22.5
• Section 59.23: Examples of sheaves
  • Definition 59.23.1
  • Remark 59.23.2
  • Definition 59.23.3
  • Remark 59.23.4
• Section 59.24: Picard groups
  • Theorem 59.24.1
• Section 59.25: The étale site
• Section 59.26: Étale morphisms
  • Definition 59.26.1
  • Proposition 59.26.2
  • Definition 59.26.3
  • Theorem 59.26.4
• Section 59.27: Étale coverings
  • Definition 59.27.1
  • Lemma 59.27.2
  • Definition 59.27.3
  • Proposition 59.27.4
• Section 59.28: Kummer theory
  • Lemma 59.28.1
  • Remark 59.28.2
  • Lemma 59.28.3
  • Remark 59.28.4
    • Equation 59.28.4.1
  • Lemma 59.28.5
• Section 59.29: Neighborhoods, stalks and points
  • Definition 59.29.1
  • Remark 59.29.2
  • Remark 59.29.3
  • Lemma 59.29.4
  • Lemma 59.29.5
  • Definition 59.29.6
  • Lemma 59.29.7
  • Remark 59.29.8
  • Lemma 59.29.9
  • Theorem 59.29.10
  • Remarks 59.29.11
  • Lemma 59.29.12
• Section 59.30: Points in other topologies
  • Lemma 59.30.1
  • Remark 59.30.2 reference
• Section 59.31: Supports of abelian sheaves
  • Lemma 59.31.1
  • Lemma 59.31.2
  • Definition 59.31.3
  • Lemma 59.31.4
  • Lemma 59.31.5
• Section 59.32: Henselian rings
  • Theorem 59.32.1
  • Definition 59.32.2
  • Theorem 59.32.3
  • Theorem 59.32.4
  • Lemma 59.32.5
  • Definition 59.32.6
  • Example 59.32.7
  • Theorem 59.32.8
• Section 59.33: Stalks of the structure sheaf
  • Lemma 59.33.1 slogan
  • Definition 59.33.2
  • Lemma 59.33.3
  • Remark 59.33.4
  • Lemma 59.33.5
• Section 59.34: Functoriality of small étale topos
  • Equation 59.34.0.1
• Section 59.35: Direct images
  • Definition 59.35.1
  • Remark 59.35.2
  • Definition 59.35.3
  • Definition 59.35.4
• Section 59.36: Inverse image
  • Definition 59.36.1
    • Equation 59.36.1.1
  • Lemma 59.36.2
  • Remark 59.36.3
• Section 59.37: Functoriality of big topoi
• Section 59.38: Functoriality and sheaves of modules
• Section 59.39: Comparing topologies
  • Lemma 59.39.1
  • Lemma 59.39.2
  • Lemma 59.39.3
  • Lemma 59.39.4
  • Lemma 59.39.5
• Section 59.40: Recovering morphisms
  • Lemma 59.40.1
  • Lemma 59.40.2
  • Lemma 59.40.3
  • Lemma 59.40.4
  • Theorem 59.40.5
• Section 59.41: Push and pull
• Section 59.42: Property (A)
  • Lemma 59.42.1
  • Lemma 59.42.2
  • Lemma 59.42.3
  • Lemma 59.42.4
• Section 59.43: Property (B)
  • Lemma 59.43.1
  • Lemma 59.43.2
  • Lemma 59.43.3
  • Lemma 59.43.4
  • Lemma 59.43.5
• Section 59.44: Property (C)
  • Lemma 59.44.1
  • Remark 59.44.2
  • Lemma 59.44.3
  • Lemma 59.44.4
  • Lemma 59.44.5
• Section 59.45: Topological invariance of the small étale site
  • Theorem 59.45.1
    • Equation 59.45.1.1
    • Equation 59.45.1.2
  • Theorem 59.45.2 reference
    • Equation 59.45.2.1
  • Remark 59.45.3
  • Proposition 59.45.4: Topological invariance of étale cohomology
• Section 59.46: Closed immersions and pushforward
  • Lemma 59.46.1
  • Lemma 59.46.2
  • Lemma 59.46.3
  • Proposition 59.46.4
• Section 59.47: Integral universally injective morphisms
  • Proposition 59.47.1
• Section 59.48: Big sites and pushforward
  • Lemma 59.48.1
  • Remark 59.48.2
  • Lemma 59.48.3
  • Lemma 59.48.4
  • Lemma 59.48.5
  • Lemma 59.48.6
  • Remark 59.48.7
• Section 59.49: Exactness of big lower shriek
  • Lemma 59.49.1
  • Lemma 59.49.2
• Section 59.50: Étale cohomology
  • Lemma 59.50.1: Mayer-Vietoris for étale cohomology
  • Lemma 59.50.2: Relative Mayer-Vietoris
• Section 59.51: Colimits
  • Definition 59.51.1
  • Lemma 59.51.2
  • Theorem 59.51.3
  • Lemma 59.51.4
  • Lemma 59.51.5
  • Lemma 59.51.6
  • Lemma 59.51.7
  • Lemma 59.51.8
  • Lemma 59.51.9
  • Lemma 59.51.10
• Section 59.52: Colimits and complexes
  • Lemma 59.52.1
  • Lemma 59.52.2
  • Lemma 59.52.3
  • Lemma 59.52.4
  • Lemma 59.52.5
  • Lemma 59.52.6
• Section 59.53: Stalks of higher direct images
  • Theorem 59.53.1
  • Remark 59.53.2
• Section 59.54: The Leray spectral sequence
  • Lemma 59.54.1
  • Proposition 59.54.2: Leray spectral sequence
• Section 59.55: Vanishing of finite higher direct images
  • Lemma 59.55.1
  • Proposition 59.55.2
  • Lemma 59.55.3
  • Lemma 59.55.4
  • Lemma 59.55.5
  • Remark 59.55.6
• Section 59.56: Galois action on stalks
  • Equation 59.56.0.1
  • Definition 59.56.1
  • Example 59.56.2
  • Theorem 59.56.3
  • Remark 59.56.4
  • Lemma 59.56.5
  • Remark 59.56.6
• Section 59.57: Group cohomology
  • Definition 59.57.1
  • Definition 59.57.2
  • Lemma 59.57.3
  • Lemma 59.57.4
  • Lemma 59.57.5
  • Lemma 59.57.6
• Section 59.58: Tate's continuous cohomology
• Section 59.59: Cohomology of a point
  • Lemma 59.59.1
  • Lemma 59.59.2
  • Example 59.59.3
  • Lemma 59.59.4
• Section 59.60: Cohomology of curves
• Section 59.61: Brauer groups
  • Theorem 59.61.1
  • Lemma 59.61.2
  • Definition 59.61.3
  • Definition 59.61.4
  • Lemma 59.61.5 slogan
    • Equation 59.61.5.1
  • Theorem 59.61.6
• Section 59.62: The Brauer group of a scheme
  • Lemma 59.62.1
  • Lemma 59.62.2 reference
• Section 59.63: The Artin-Schreier sequence
  • Lemma 59.63.1
  • Lemma 59.63.2
  • Lemma 59.63.3
  • Lemma 59.63.4
  • Lemma 59.63.5
• Section 59.64: Locally constant sheaves
  • Definition 59.64.1
  • Lemma 59.64.2
  • Lemma 59.64.3
  • Lemma 59.64.4
  • Lemma 59.64.5
  • Lemma 59.64.6
  • Lemma 59.64.7
  • Lemma 59.64.8
• Section 59.65: Locally constant sheaves and the fundamental group
  • Lemma 59.65.1
  • Lemma 59.65.2
  • Remark 59.65.3
• Section 59.66: Méthode de la trace
  • Definition 59.66.1
  • Lemma 59.66.2
  • Lemma 59.66.3
  • Lemma 59.66.4
• Section 59.67: Galois cohomology
  • Lemma 59.67.1
  • Lemma 59.67.2
  • Lemma 59.67.3
  • Proposition 59.67.4 reference
  • Definition 59.67.5
  • Example 59.67.6
  • Lemma 59.67.7
  • Theorem 59.67.8
  • Definition 59.67.9
  • Theorem 59.67.10: Tsen's theorem
  • Lemma 59.67.11
  • Lemma 59.67.12
• Section 59.68: Higher vanishing for the multiplicative group
  • Theorem 59.68.1: The Fundamental Exact Sequence
  • Lemma 59.68.2
  • Lemma 59.68.3
  • Lemma 59.68.4
  • Theorem 59.68.5
• Section 59.69: Picard groups of curves
  • Lemma 59.69.1
  • Lemma 59.69.2
  • Lemma 59.69.3
  • Remark 59.69.4
  • Remark 59.69.5
• Section 59.70: Extension by zero
  • Definition 59.70.1
    • Equation 59.70.1.1
  • Exercise 59.70.2
  • Proposition 59.70.3
  • Lemma 59.70.4
  • Lemma 59.70.5: Extension by zero commutes with base change
  • Lemma 59.70.6
  • Lemma 59.70.7
  • Lemma 59.70.8
  • Lemma 59.70.9
• Section 59.71: Constructible sheaves
  • Definition 59.71.1
  • Lemma 59.71.2
  • Lemma 59.71.3
  • Lemma 59.71.4
  • Lemma 59.71.5
  • Lemma 59.71.6
  • Lemma 59.71.7
  • Lemma 59.71.8
  • Lemma 59.71.9
  • Lemma 59.71.10
• Section 59.72: Auxiliary lemmas on morphisms
  • Lemma 59.72.1
  • Lemma 59.72.2
  • Lemma 59.72.3
• Section 59.73: More on constructible sheaves
  • Lemma 59.73.1
  • Lemma 59.73.2
  • Lemma 59.73.3
  • Lemma 59.73.4
  • Lemma 59.73.5
  • Lemma 59.73.6
  • Lemma 59.73.7
  • Lemma 59.73.8
  • Lemma 59.73.9
  • Lemma 59.73.10
  • Lemma 59.73.11
  • Lemma 59.73.12
  • Lemma 59.73.13
  • Lemma 59.73.14
• Section 59.74: Constructible sheaves on Noetherian schemes
  • Proposition 59.74.1
  • Lemma 59.74.2
  • Lemma 59.74.3
  • Lemma 59.74.4 reference
  • Lemma 59.74.5
• Section 59.75: Specializations and étale sheaves
  • Remark 59.75.1: Alternative description of sp
  • Remark 59.75.2: Yet another description of sp
  • Remark 59.75.3: Lifting specializations
  • Lemma 59.75.4
  • Lemma 59.75.5
  • Lemma 59.75.6
  • Lemma 59.75.7
  • Remark 59.75.8
• Section 59.76: Complexes with constructible cohomology
  • Definition 59.76.1
  • Lemma 59.76.2
  • Lemma 59.76.3
  • Lemma 59.76.4
  • Lemma 59.76.5
  • Lemma 59.76.6
• Section 59.77: Tor finite with constructible cohomology
  • Definition 59.77.1
  • Remark 59.77.2
  • Lemma 59.77.3
  • Remark 59.77.4
  • Lemma 59.77.5
  • Lemma 59.77.6
  • Lemma 59.77.7
• Section 59.78: Torsion sheaves
  • Lemma 59.78.1
  • Lemma 59.78.2
• Section 59.79: Cohomology with support in a closed subscheme
  • Lemma 59.79.1
  • Lemma 59.79.2
  • Lemma 59.79.3
  • Lemma 59.79.4
  • Lemma 59.79.5
• Section 59.80: Schemes with strictly henselian local rings
  • Lemma 59.80.1
  • Lemma 59.80.2
  • Lemma 59.80.3
  • Lemma 59.80.4
  • Lemma 59.80.5
  • Lemma 59.80.6
  • Lemma 59.80.7
  • Lemma 59.80.8
  • Lemma 59.80.9
• Section 59.81: Absolutely integrally closed vanishing
  • Lemma 59.81.1
  • Lemma 59.81.2
    • Equation 59.81.2.1
  • Lemma 59.81.3
  • Lemma 59.81.4
  • Proposition 59.81.5
• Section 59.82: Affine analog of proper base change
  • Lemma 59.82.1
  • Lemma 59.82.2
  • Lemma 59.82.3
  • Lemma 59.82.4
  • Example 59.82.5
  • Lemma 59.82.6
  • Theorem 59.82.7: Gabber
  • Lemma 59.82.8
• Section 59.83: Cohomology of torsion sheaves on curves
  • Situation 59.83.1
  • Lemma 59.83.2
  • Remark 59.83.3: Invariance under extension of algebraically closed ground field
  • Lemma 59.83.4
  • Lemma 59.83.5
  • Lemma 59.83.6
  • Lemma 59.83.7
  • Lemma 59.83.8
  • Lemma 59.83.9
  • Theorem 59.83.10
  • Theorem 59.83.11
  • Lemma 59.83.12
• Section 59.84: Cohomology of torsion modules on curves
  • Lemma 59.84.1
  • Lemma 59.84.2
  • Lemma 59.84.3
  • Lemma 59.84.4
  • Lemma 59.84.5
  • Lemma 59.84.6
  • Theorem 59.84.7
• Section 59.85: First cohomology of proper schemes
  • Lemma 59.85.1
  • Lemma 59.85.2
• Section 59.86: Preliminaries on base change
  • Lemma 59.86.1
  • Lemma 59.86.2
  • Lemma 59.86.3
  • Lemma 59.86.4
  • Lemma 59.86.5
  • Lemma 59.86.6
• Section 59.87: Base change for pushforward
  • Lemma 59.87.1
  • Lemma 59.87.2
  • Lemma 59.87.3
  • Lemma 59.87.4
  • Lemma 59.87.5
  • Lemma 59.87.6
• Section 59.88: Base change for higher direct images
  • Remark 59.88.1
  • Lemma 59.88.2
  • Lemma 59.88.3
  • Lemma 59.88.4
  • Lemma 59.88.5
  • Lemma 59.88.6
  • Lemma 59.88.7
• Section 59.89: Smooth base change
  • Lemma 59.89.1
  • Theorem 59.89.2: Smooth base change
  • Lemma 59.89.3
• Section 59.90: Applications of smooth base change
  • Lemma 59.90.1
  • Lemma 59.90.2
  • Lemma 59.90.3
• Section 59.91: The proper base change theorem
  • Lemma 59.91.1
  • Lemma 59.91.2
  • Lemma 59.91.3
  • Lemma 59.91.4
  • Lemma 59.91.5
    • Equation 59.91.5.1
    • Equation 59.91.5.2
    • Equation 59.91.5.3
  • Lemma 59.91.6
  • Lemma 59.91.7
  • Lemma 59.91.8
  • Lemma 59.91.9 slogan
  • Lemma 59.91.10
  • Theorem 59.91.11
  • Lemma 59.91.12
  • Lemma 59.91.13
• Section 59.92: Applications of proper base change
  • Lemma 59.92.1
  • Lemma 59.92.2
  • Lemma 59.92.3
  • Lemma 59.92.4
  • Lemma 59.92.5
• Section 59.93: Local acyclicity
  • Equation 59.93.0.1
  • Definition 59.93.1 reference
  • Proposition 59.93.2
  • Lemma 59.93.3
  • Lemma 59.93.4
• Section 59.94: The cospecialization map
  • Lemma 59.94.1
  • Lemma 59.94.2
  • Lemma 59.94.3
  • Lemma 59.94.4
• Section 59.95: Cohomological dimension
  • Definition 59.95.1
  • Lemma 59.95.2
  • Lemma 59.95.3
  • Lemma 59.95.4
  • Lemma 59.95.5
  • Proposition 59.95.6
  • Lemma 59.95.7
  • Lemma 59.95.8
• Section 59.96: Finite cohomological dimension
  • Definition 59.96.1
  • Lemma 59.96.2
  • Lemma 59.96.3
  • Lemma 59.96.4
  • Lemma 59.96.5
  • Lemma 59.96.6
• Section 59.97: Künneth in étale cohomology
  • Remark 59.97.1
  • Lemma 59.97.2
  • Lemma 59.97.3
  • Lemma 59.97.4
  • Lemma 59.97.5
  • Lemma 59.97.6
  • Lemma 59.97.7
  • Lemma 59.97.8
  • Lemma 59.97.9
• Section 59.98: Comparing chaotic and Zariski topologies
  • Lemma 59.98.1
• Section 59.99: Comparing big and small topoi
  • Lemma 59.99.1
    • Equation 59.99.1.1
  • Lemma 59.99.2
  • Lemma 59.99.3
  • Lemma 59.99.4
  • Lemma 59.99.5
• Section 59.100: Comparing fppf and étale topologies
  • Lemma 59.100.1
  • Lemma 59.100.2
  • Lemma 59.100.3
  • Lemma 59.100.4
  • Lemma 59.100.5
  • Lemma 59.100.6
  • Lemma 59.100.7
  • Lemma 59.100.8
• Section 59.101: Comparing fppf and étale topologies: modules
  • Lemma 59.101.1
  • Lemma 59.101.2
  • Lemma 59.101.3
• Section 59.102: Comparing ph and étale topologies
  • Lemma 59.102.1
  • Lemma 59.102.2
  • Lemma 59.102.3
  • Lemma 59.102.4
  • Lemma 59.102.5
  • Lemma 59.102.6
  • Lemma 59.102.7
• Section 59.103: Comparing h and étale topologies
  • Lemma 59.103.1
  • Lemma 59.103.2
  • Lemma 59.103.3
  • Lemma 59.103.4
  • Lemma 59.103.5
  • Lemma 59.103.6
  • Lemma 59.103.7
• Section 59.104: Descending étale sheaves
  • Lemma 59.104.1
  • Lemma 59.104.2
  • Lemma 59.104.3
  • Lemma 59.104.4
  • Lemma 59.104.5
  • Lemma 59.104.6
  • Lemma 59.104.7
  • Lemma 59.104.8
• Section 59.105: Blow up squares and étale cohomology
  • Lemma 59.105.1
  • Lemma 59.105.2
  • Lemma 59.105.3
• Section 59.106: Almost blow up squares and the h topology
  • Equation 59.106.0.1
  • Lemma 59.106.1
  • Proposition 59.106.2
  • Lemma 59.106.3
• Section 59.107: Cohomology of the structure sheaf in the h topology
  • Lemma 59.107.1
  • Proposition 59.107.2