35 Descent

Section 35.1: Introduction
Section 35.2: Descent data for quasi-coherent sheaves
- Definition 35.2.1
- Lemma 35.2.2
- Definition 35.2.3
- Lemma 35.2.4
Section 35.3: Descent for modules
- Definition 35.3.1
- Lemma 35.3.2
- Lemma 35.3.3
- Definition 35.3.4
  - Equation 35.3.4.1
- Lemma 35.3.5
- Lemma 35.3.6
- Lemma 35.3.7
- Lemma 35.3.8
- Proposition 35.3.9 slogan
- Remark 35.3.10
- Remark 35.3.11
Section 35.4: Descent for universally injective morphisms
- Subsection 35.4.1: Category-theoretic preliminaries
  - Equation 35.4.1.1
- Definition 35.4.2
  - Equation 35.4.2.1
- Subsection 35.4.3: Universally injective morphisms
- Remark 35.4.4
- Definition 35.4.5
- Example 35.4.6
- Example 35.4.7
- Lemma 35.4.8
- Definition 35.4.9
- Lemma 35.4.10
- Remark 35.4.11
  - Equation 35.4.11.1
- Lemma 35.4.12
- Remark 35.4.13
- Subsection 35.4.14: Descent for modules and their morphisms
  - Equation 35.4.14.1
- Definition 35.4.15
- Lemma 35.4.16
  - Equation 35.4.16.1
- Lemma 35.4.17
  - Equation 35.4.17.1
- Lemma 35.4.18
  - Equation 35.4.18.1
- Definition 35.4.19
  - Equation 35.4.19.1
- Lemma 35.4.20
  - Equation 35.4.20.1
- Remark 35.4.21
- Theorem 35.4.22
- Subsection 35.4.23: Descent for properties of modules
- Lemma 35.4.24
- Theorem 35.4.25
- Theorem 35.4.26
- Remark 35.4.27
Section 35.5: Fpqc descent of quasi-coherent sheaves
- Lemma 35.5.1
- Proposition 35.5.2
Section 35.6: Galois descent for quasi-coherent sheaves
- Lemma 35.6.1
- Lemma 35.6.2
Section 35.7: Descent of finiteness properties of modules
- Lemma 35.7.1
- Lemma 35.7.2
- Lemma 35.7.3
- Lemma 35.7.4
- Lemma 35.7.5
- Lemma 35.7.6
- Lemma 35.7.7
- Remark 35.7.8
- Lemma 35.7.9
- Lemma 35.7.10
Section 35.8: Quasi-coherent sheaves and topologies, I
- Equation 35.8.0.1
- Lemma 35.8.1
- Definition 35.8.2
- Remark 35.8.3
- Remark 35.8.4
- Lemma 35.8.5
- Remark 35.8.6
- Lemma 35.8.7
- Lemma 35.8.8
- Proposition 35.8.9
- Lemma 35.8.10
Section 35.9: Cohomology of quasi-coherent modules and topologies
- Lemma 35.9.1
- Lemma 35.9.2
- Proposition 35.9.3 slogan
- Proposition 35.9.4
- Lemma 35.9.5
Section 35.10: Quasi-coherent sheaves and topologies, II
- Lemma 35.10.1
- Lemma 35.10.2
- Lemma 35.10.3
- Lemma 35.10.4
- Example 35.10.5
- Lemma 35.10.6
Section 35.11: Quasi-coherent modules and affines
- Equation 35.11.0.1
- Lemma 35.11.1
  - Equation 35.11.1.1
- Lemma 35.11.2
Section 35.12: Parasitic modules
- Definition 35.12.1
- Lemma 35.12.2
- Lemma 35.12.3
- Lemma 35.12.4
Section 35.13: Fpqc coverings are universal effective epimorphisms
- Lemma 35.13.1
- Lemma 35.13.2
- Lemma 35.13.3
- Lemma 35.13.4
- Lemma 35.13.5
- Lemma 35.13.6
- Lemma 35.13.7
- Lemma 35.13.8
Section 35.14: Descent of finiteness and smoothness properties of morphisms
- Lemma 35.14.1
- Lemma 35.14.2
- Lemma 35.14.3 reference
- Lemma 35.14.4
- Lemma 35.14.5
- Remark 35.14.6
- Lemma 35.14.7
- Lemma 35.14.8
Section 35.15: Local properties of schemes
- Definition 35.15.1
- Lemma 35.15.2
- Remark 35.15.3
Section 35.16: Properties of schemes local in the fppf topology
- Lemma 35.16.1
- Lemma 35.16.2
- Lemma 35.16.3
Section 35.17: Properties of schemes local in the syntomic topology
- Lemma 35.17.1
- Lemma 35.17.2
Section 35.18: Properties of schemes local in the smooth topology
- Lemma 35.18.1
- Lemma 35.18.2 slogan
- Lemma 35.18.3
- Lemma 35.18.4
- Lemma 35.18.5
Section 35.19: Variants on descending properties
- Lemma 35.19.1
- Lemma 35.19.2
Section 35.20: Germs of schemes
- Definition 35.20.1
- Definition 35.20.2
Section 35.21: Local properties of germs
- Definition 35.21.1
- Lemma 35.21.2
- Lemma 35.21.3
- Lemma 35.21.4
Section 35.22: Properties of morphisms local on the target
- Definition 35.22.1
- Lemma 35.22.2
- Lemma 35.22.3
- Lemma 35.22.4
- Remark 35.22.5
Section 35.23: Properties of morphisms local in the fpqc topology on the target
- Lemma 35.23.1
- Lemma 35.23.2
- Lemma 35.23.3
- Lemma 35.23.4
- Lemma 35.23.5
- Lemma 35.23.6
- Lemma 35.23.7
- Lemma 35.23.8
- Lemma 35.23.9
- Lemma 35.23.10
- Lemma 35.23.11
- Lemma 35.23.12
- Lemma 35.23.13
- Lemma 35.23.14
- Lemma 35.23.15
- Lemma 35.23.16
- Lemma 35.23.17
- Lemma 35.23.18
- Lemma 35.23.19
- Lemma 35.23.20
- Lemma 35.23.21
- Lemma 35.23.22
- Lemma 35.23.23
- Lemma 35.23.24
- Lemma 35.23.25
- Lemma 35.23.26
- Lemma 35.23.27
- Lemma 35.23.28
- Lemma 35.23.29
- Lemma 35.23.30
- Lemma 35.23.31
- Lemma 35.23.32
Section 35.24: Properties of morphisms local in the fppf topology on the target
- Lemma 35.24.1
Section 35.25: Application of fpqc descent of properties of morphisms
- Lemma 35.25.1
- Lemma 35.25.2
- Lemma 35.25.3
- Lemma 35.25.4
- Lemma 35.25.5
- Lemma 35.25.6
Section 35.26: Properties of morphisms local on the source
- Definition 35.26.1
- Lemma 35.26.2
- Lemma 35.26.3
- Lemma 35.26.4
- Remark 35.26.5
Section 35.27: Properties of morphisms local in the fpqc topology on the source
- Lemma 35.27.1
- Lemma 35.27.2
Section 35.28: Properties of morphisms local in the fppf topology on the source
- Lemma 35.28.1
- Lemma 35.28.2
- Lemma 35.28.3
- Lemma 35.28.4
Section 35.29: Properties of morphisms local in the syntomic topology on the source
- Lemma 35.29.1
Section 35.30: Properties of morphisms local in the smooth topology on the source
- Lemma 35.30.1
Section 35.31: Properties of morphisms local in the étale topology on the source
- Lemma 35.31.1
- Lemma 35.31.2
- Lemma 35.31.3
Section 35.32: Properties of morphisms étale local on source-and-target
- Example 35.32.1
- Example 35.32.2
- Definition 35.32.3
- Lemma 35.32.4
- Lemma 35.32.5
- Lemma 35.32.6
- Remark 35.32.7
- Remark 35.32.8
- Lemma 35.32.9
- Lemma 35.32.10
- Lemma 35.32.11
Section 35.33: Properties of morphisms of germs local on source-and-target
- Definition 35.33.1
- Lemma 35.33.2
- Lemma 35.33.3
- Lemma 35.33.4
- Lemma 35.33.5
- Lemma 35.33.6
- Lemma 35.33.7
- Lemma 35.33.8
Section 35.34: Descent data for schemes over schemes
- Definition 35.34.1
- Remark 35.34.2
- Definition 35.34.3
- Remark 35.34.4
- Lemma 35.34.5
- Lemma 35.34.6
- Definition 35.34.7
- Lemma 35.34.8: Pullback of descent data for schemes over families
- Definition 35.34.9
- Definition 35.34.10
- Definition 35.34.11
Section 35.35: Fully faithfulness of the pullback functors
- Lemma 35.35.1
- Lemma 35.35.2
- Lemma 35.35.3
- Lemma 35.35.4
- Lemma 35.35.5
- Lemma 35.35.6
- Lemma 35.35.7
- Lemma 35.35.8
- Lemma 35.35.9
- Lemma 35.35.10
- Lemma 35.35.11
- Remark 35.35.12
- Lemma 35.35.13
Section 35.36: Descending types of morphisms
- Definition 35.36.1
- Lemma 35.36.2
Section 35.37: Descending affine morphisms
- Lemma 35.37.1
- Lemma 35.37.2
Section 35.38: Descending quasi-affine morphisms
- Lemma 35.38.1
Section 35.39: Descent data in terms of sheaves
- Lemma 35.39.1
- Remark 35.39.2