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6. Galois Descent for Quasi-Coherent Sheaves 19 7 DESCENT 0238 Contents 1. Introduction 2 2. Descent data for quasi-coherent sheaves 2 3. Descent for modules 4 4. Descent for universally injective morphisms 9 4.1. Category-theoretic preliminaries 10 4.3. Universally injective morphisms 11 4.14. Descent for modules and their morphisms 12 4.23. Descent for properties of modules 15 5. Fpqc descent of quasi-coherent sheaves 17 6. Galois descent for quasi-coherent sheaves 19 7. Descent of finiteness properties of modules 21 8. Quasi-coherent sheaves and topologies 23 9. Parasitic modules 31 10. Fpqc coverings are universal effective epimorphisms 32 11. Descent of finiteness and smoothness properties of morphisms 36 12. Local properties of schemes 39 13. Properties of schemes local in the fppf topology 40 14. Properties of schemes local in the syntomic topology 42 15. Properties of schemes local in the smooth topology 42 16. Variants on descending properties 43 17. Germs of schemes 44 18. Local properties of germs 44 19. Properties of morphisms local on the target 45 20. Properties of morphisms local in the fpqc topology on the target 47 21. Properties of morphisms local in the fppf topology on the target 54 22. Application of fpqc descent of properties of morphisms 54 23. Properties of morphisms local on the source 57 24. Properties of morphisms local in the fpqc topology on the source 58 25. Properties of morphisms local in the fppf topology on the source 58 26. Properties of morphisms local in the syntomic topology on the source 59 27. Properties of morphisms local in the smooth topology on the source 59 28. Properties of morphisms local in the étale topology on the source 60 29. Properties of morphisms étale local on source-and-target 60 30. Properties of morphisms of germs local on source-and-target 69 31. Descent data for schemes over schemes 71 32. Fully faithfulness of the pullback functors 75 33. Descending types of morphisms 80 34. Descending affine morphisms 82 This is a chapter of the Stacks Project, version fac02ecd, compiled on Sep 14, 2021. 1 DESCENT 2 35. Descending quasi-affine morphisms 82 36. Descent data in terms of sheaves 83 37. Other chapters 84 References 86 1. Introduction 0239 In the chapter on topologies on schemes (see Topologies, Section 1) we introduced Zariski, étale, fppf, smooth, syntomic and fpqc coverings of schemes. In this chapter we discuss what kind of structures over schemes can be descended through such coverings. See for example [Gro95a], [Gro95b], [Gro95e], [Gro95f], [Gro95c], and [Gro95d]. This is also meant to introduce the notions of descent, descent data, effective descent data, in the less formal setting of descent questions for quasi- coherent sheaves, schemes, etc. The formal notion, that of a stack over a site, is discussed in the chapter on stacks (see Stacks, Section 1). 2. Descent data for quasi-coherent sheaves 023A In this chapter we will use the convention where the projection maps pri : X ×...× X → X are labeled starting with i = 0. Hence we have pr0, pr1 : X × X → X, pr0, pr1, pr2 : X × X × X → X, etc. 023BDefinition 2.1. Let S be a scheme. Let {fi : Si → S}i∈I be a family of morphisms with target S. (1) A descent datum (Fi, ϕij) for quasi-coherent sheaves with respect to the given family is given by a quasi-coherent sheaf Fi on Si for each i ∈ I, an ∗ ∗ isomorphism of quasi-coherent OSi×S Sj -modules ϕij : pr0Fi → pr1Fj for each pair (i, j) ∈ I2 such that for every triple of indices (i, j, k) ∈ I3 the diagram ∗ ∗ pr Fi / pr Fk 0 pr∗ ϕ 2 02 ik : ∗ ∗ pr01ϕij pr12ϕjk $ ∗ pr1Fj of OSi×S Sj ×S Sk -modules commutes. This is called the cocycle condition. 0 0 (2) A morphism ψ :(Fi, ϕij) → (Fi , ϕij) of descent data is given by a family 0 ψ = (ψi)i∈I of morphisms of OSi -modules ψi : Fi → Fi such that all the diagrams ∗ ∗ pr Fi / pr Fj 0 ϕij 1 ∗ ∗ pr0 ψi pr1 ψj ϕ0 ∗ 0 ij ∗ 0 pr0Fi / pr1Fj commute. S A good example to keep in mind is the following. Suppose that S = Si is an open covering. In that case we have seen descent data for sheaves of sets in Sheaves, Section 33 where we called them “glueing data for sheaves of sets with respect to the given covering”. Moreover, we proved that the category of glueing data is equivalent DESCENT 3 to the category of sheaves on S. We will show the analogue in the setting above when {Si → S}i∈I is an fpqc covering. In the extreme case where the covering {S → S} is given by idS a descent datum is necessarily of the form (F, idF ). The cocycle condition guarantees that the identity on F is the only permitted map in this case. The following lemma shows in particular that to every quasi-coherent sheaf of OS-modules there is associated a unique descent datum with respect to any given family. 023C Lemma 2.2. Let U = {Ui → U}i∈I and V = {Vj → V }j∈J be families of morphisms of schemes with fixed target. Let (g, α : I → J, (gi)) : U → V be a morphism of families of maps with fixed target, see Sites, Definition 8.1. Let (Fj, ϕjj0 ) be a descent datum for quasi-coherent sheaves with respect to the family {Vj → V }j∈J . Then (1) The system ∗ ∗ gi Fα(i), (gi × gi0 ) ϕα(i)α(i0) is a descent datum with respect to the family {Ui → U}i∈I . (2) This construction is functorial in the descent datum (Fj, ϕjj0 ). 0 0 0 (3) Given a second morphism (g , α : I → J, (gi)) of families of maps with fixed target with g = g0 there exists a functorial isomorphism of descent data ∗ ∗ ∼ 0 ∗ 0 0 ∗ (gi Fα(i), (gi × gi0 ) ϕα(i)α(i0)) = ((gi) Fα0(i), (gi × gi0 ) ϕα0(i)α0(i0)). ∗ 0 ∗ Proof. Omitted. Hint: The maps gi Fα(i) → (gi) Fα0(i) which give the isomor- phism of descent data in part (3) are the pullbacks of the maps ϕα(i)α0(i) by the 0 morphisms (gi, gi): Ui → Vα(i) ×V Vα0(i). Any family U = {Si → S}i∈I is a refinement of the trivial covering {S → S} in a unique way. For a quasi-coherent sheaf F on S we denote simply (F|Si , can) the descent datum with respect to U obtained by the procedure above. 023DDefinition 2.3. Let S be a scheme. Let {Si → S}i∈I be a family of morphisms with target S. (1) Let F be a quasi-coherent OS-module. We call the unique descent on F datum with respect to the covering {S → S} the trivial descent datum. (2) The pullback of the trivial descent datum to {Si → S} is called the canon- ical descent datum. Notation: (F|Si , can). (3) A descent datum (Fi, ϕij) for quasi-coherent sheaves with respect to the given covering is said to be effective if there exists a quasi-coherent sheaf F on S such that (Fi, ϕij) is isomorphic to (F|Si , can). S 023E Lemma 2.4. Let S be a scheme. Let S = Ui be an open covering. Any descent datum on quasi-coherent sheaves for the family U = {Ui → S} is effective. More- over, the functor from the category of quasi-coherent OS-modules to the category of descent data with respect to U is fully faithful. Proof. This follows immediately from Sheaves, Section 33 and the fact that being quasi-coherent is a local property, see Modules, Definition 10.1. To prove more we first need to study the case of modules over rings. DESCENT 4 3. Descent for modules 023F Let R → A be a ring map. By Simplicial, Example 5.5 this gives rise to a cosim- plicial R-algebra / o / A o A ⊗R A / A ⊗R A ⊗R A / o / Let us denote this (A/R)• so that (A/R)n is the (n + 1)-fold tensor product of A over R. Given a map ϕ :[n] → [m] the R-algebra map (A/R)•(ϕ) is the map Y Y Y a0 ⊗ ... ⊗ an 7−→ ai ⊗ ai ⊗ ... ⊗ ai ϕ(i)=0 ϕ(i)=1 ϕ(i)=m where we use the convention that the empty product is 1. Thus the first few maps, notation as in Simplicial, Section 5, are 1 δ0 : a0 7→ 1 ⊗ a0 1 δ1 : a0 7→ a0 ⊗ 1 0 σ0 : a0 ⊗ a1 7→ a0a1 2 δ0 : a0 ⊗ a1 7→ 1 ⊗ a0 ⊗ a1 2 δ1 : a0 ⊗ a1 7→ a0 ⊗ 1 ⊗ a1 2 δ2 : a0 ⊗ a1 7→ a0 ⊗ a1 ⊗ 1 1 σ0 : a0 ⊗ a1 ⊗ a2 7→ a0a1 ⊗ a2 1 σ1 : a0 ⊗ a1 ⊗ a2 7→ a0 ⊗ a1a2 and so on. An R-module M gives rise to a cosimplicial (A/R)•-module (A/R)• ⊗R M. In other words Mn = (A/R)n ⊗R M and using the R-algebra maps (A/R)n → (A/R)m to define the corresponding maps on M ⊗R (A/R)•. The analogue to a descent datum for quasi-coherent sheaves in the setting of mod- ules is the following. 023GDefinition 3.1. Let R → A be a ring map. (1) A descent datum (N, ϕ) for modules with respect to R → A is given by an A-module N and an isomorphism of A ⊗R A-modules ϕ : N ⊗R A → A ⊗R N such that the cocycle condition holds: the diagram of A ⊗R A ⊗R A-module maps N ⊗R A ⊗R A / A ⊗R A ⊗R N ϕ02 6 ϕ01 ϕ12 ( A ⊗R N ⊗R A commutes (see below for notation).
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