                                               Source file: src/flat.tex

FLAT DESCENT FOR QUASI-COHERENT SHEAVES

Contents

Section 1. Introduction 1 Subsection 1.1. Descent for quasi-coherent sheaves 1 References 5

Section 1. Introduction

In this chapter we discuss the flat topology.

Subsection 1.1. Descent for quasi-coherent sheaves. Let me state our goal now so we have a clear idea of what we are trying to achieve. It is to show that the fibered category (QCoh/S) (of quasi-coherent sheaves over a scheme S) over Sch/S (the category of schemes over S) is a stack with respect to the flat topology. The idea is that we will be able to exploit the fact that there is a standard equivalence of categories between QCoh(U) and Mod_{A} where U = Spec(A), and then use Lemma 1.1.1.

Lemma 1.1.1. ([Vis],Lemma 4.25) Let S be a scheme, F be a fibered category over the category (Sch/S). Suppose the following conditions are satisfied.

(1) F is a stack with respect to the Zariski topology.

(2) Whenever V → U is a flat surjective morphism of affine S-schemes, the

functor F(U) → F(V → U) is an equivalence of categories. Then F is a stack with respect to the flat (fpqc) topology.

Proof. FIXME: Add proof. □

Note that the fpqc topology stands for ”fidelment plat et quasi-compact” which means faithfully flat and quasi-compact. This is a finer topology than the fppf topology which is a finer topology than the etale topology. In the case of (QCoh/S) over (Sch/S) we can easily see that the first condition is satisfied (i.e in the Zariski topology our definition for quasi-coherent sheaves is exactly the conditions needed for the descent datum to be effective). For the second condition, it will be necessary to define certain notions and to prove certain algebraic results. We begin with some definitions.

Definition 1.1.2. A morphism of schemes f : X → Y is faithfully flat if it is flat and surjective. Let B be an algebra over A, we say that B is faithfully flat if the associated morphism of schemes SpecB → SpecA is.

Proposition 1.1.3. Let B be an algebra over A, the following are equivalent. (1) B is faithfully flat over A.

(2) A sequence of A-modules M′ → M → M′′ is exact if and only if the induced sequence of B-modules M′ ⊗_{A} B → M ⊗_{A} B → M′′ ⊗_{A} B is exact.

(3) A homomorphism of A modules M′ → M is injective if and only if the associated homomorphism of B-modules M′ ⊗_{A} B → M ⊗_{A} B is injective.

(4) B is flat over A, and if M is a module over A with M ⊗_{A} B = 0, we have M = 0.

(5) B is flat over A, and mB ≠ B for all maximal ideals m of A.

Definition 1.1.4. Let C be a site. Let F be a category fibered over C. Let U = {σ_{i} : U_{i} → U} be a covering in C. An object with descent data ({ξ_{i}}, {ϕ_{ij}}) on U is a collection of objects ξ_{i} ∈ F(U_{i}), together with isomorphisms ϕ_{ij} : pr_{2}^{∗}ξ_{j} ≃ pr_{1}^{∗}ξ_{i} in F(U_{i} ×_{U} U_{j}), such that the following cocyle condition is satisfied: for any triple of indicies i, j, k we have the equality pr_{13}^{∗}ϕ_{ik} = pr_{12}^{∗}ϕ_{ij} ∘ pr_{23}^{∗}ϕ_{jk} : pr_{3}^{∗}ξ_{k} → pr_{1}^{∗}ξ_{i}. Where pr_{ab} and pr_{a} projections onto the a^{th} and b^{th} factor, or the the a^{th} factor respectively. An arrow between objects with descent data {α_{i}} : ({ξ_{i}}, {ϕ_{ij}}) → ({η_{i}}, {ψ_{ij}}) is a collection of arrows α_{i} : ξ_{i} → η_{i} in F(U_{i}) with the property that for each pair of indicies i, j the cocycle condition is satisfied. There is an obvious way of composing morphisms, which makes the objects with descent data the objects of a category which we will denote F({U_{i} → U}) Specifically in the case of the lemma we are only interested in the case where U = Spec(A) is affine. So it can be covered by one affine scheme V = Spec(B). In this case the above definition reduces to the following: F(V → U) is the category consisting of pairs (ξ, ϕ) where ξ ∈ F(V) and ϕ : p_{1}^{∗}ξ ≃ p_{2}^{∗}ξ is an isomorphism which satisfies the cocycle condition.

To continue we need to set some conventions. Let A be a commutative ring, and denote Mod_{A} as the category of modules over A. We also have a ring homomorphism f : A → B. We define a category Mod_{A→B} as follows. Let the objects be pairs (N, ϕ) where N is a B-module and ϕ : N ⊗_{A} B ≃ B ⊗_{A} N is an isomorphism of B^{⊗p2}-modules such that the following cocycle condition is satisfied:

ϕ_{1} : B ⊗_{A} N ⊗_{A} B → B ⊗_{A} B ⊗_{A} N,
ϕ_{2} : N ⊗_{A} B ⊗_{A} B → B ⊗_{A} B ⊗_{A} B,
ϕ_{3} : N ⊗_{A} B ⊗_{A} B → B ⊗_{A} N ⊗_{A} B,

where ϕ_{1} = id_{B} ⊗ ϕ, ϕ_{3} = ϕ ⊗ id_{B}, and ϕ_{2} = ϕ_{1}ϕ_{3}. A morphism β : (N, ϕ) → (N′, ϕ′) is a homomorphism of B-modules β : N → N′ making the following diagram commute:

N ⊗_{A} B ⟶^{ϕ} B ⊗_{A} N
β ⊗ id_{B} ↓  ↓ id_{B} ⊗ β
N′ ⊗_{A} B ⟶^{ϕ′} B ⊗_{A} N′

Given a functor F : Mod_{A} → Mod_{A→B} which takes an A-module M to the pair (B ⊗_{A} M, ϕ_{M}) where ϕ_{M} : (B ⊗_{A} M) ⊗_{A} B → B ⊗_{A} (B ⊗_{A} M) maps b ⊗ m ⊗ b′ to b ⊗ b′ ⊗ m (and satisfies the cocyle condition).

Theorem 1.1.5. If B is faithfully flat over A, the functor F : Mod_{A} → Mod_{A→B} as defined above is an equivalence of categories.

To prove this we first need the following lemma Lemma 1.1.6. Let M be an A-module. Then the following sequence

0 → M ⟶^{α_{M}} B ⊗_{A} M ⟶^{(e_{1}−e_{2})⊗id_{M}} B^{⊗2} ⊗_{A} M ⟶^{(e_{1}−e_{2}+e_{3})⊗id_{M}} B^{⊗3} ⊗_{A} M → ⋯

is exact. Where e_{i} : B^{⊗n} → B^{⊗n+1} is the map that puts a 1 into the i^{th} place of the tensor product.

Proof. It is easy to see that α_{M} is an injective map, and that images of all of the maps will be contained in the appropriate kernels. So what we need to show is that we have reverse containment (i.e. kernels contained in images) at each step of the sequence. If there existed a section B ⊗_{A} M → M then this will be easy (for reasons we will see soon). However, there does not always exist such a map. Luckily we can use the fact that B is flat over A, and that a sequence of A-modules is exact if and only if the sequence tensored with B is exact. This is fortunate because once we tensor our sequence with B we get the following sequence (with the same maps as in the statement of the lemma just with tensored with a id_{B} on the left):

0 → B ⊗_{A} M → B^{⊗2} ⊗_{A} M → B^{⊗3} ⊗_{A} M → B^{⊗4} ⊗_{A} M → · · · .

And there is a natural map B^{⊗2} ⊗_{A} M →^{mult} B ⊗_{A} M which just takes b ⊗ b′ ⊗ m to the element bb′ ⊗ m. In other words it is just the multiplication map B ⊗_{A} B → B composed with the identity on M.

Now, to prove that the sequence is exact at B^{⊗2} ⊗_{A} M, we pick an element Σb_{i} ⊗ b′_{i} ⊗ m_{i} in the kernel of the map id_{B} ∘ (e_{1} − e_{2}) ∘ id_{M}. This means that in B^{⊗_{3}} ⊗_{A} M we have the following relation: Σb_{i} ⊗b′_{i} ⊗1⊗m_{i} = Σb_{i} ⊗1⊗b′_{i} ⊗m_{i}. We then apply the map mult ∘ id_{B} ∘ id_{M} to the equality and get the Σb_{i}b′_{i} ⊗ 1 ⊗ m_{i} = b_{i} ⊗ b′_{i} ⊗ m_{i} in B^{⊗2} ⊗_{A} M. So we are getting id_{B} ∘ α_{M}(Σb_{i}b′_{i} ⊗ m_{i}) = Σb_{i} ⊗ b′_{i} ⊗ m_{i} which was the element from the kernel that we chose. Thus every element in the kernel is also in the image of the appropriate map hence the sequence is exact there. The same argument can be made at each step of the sequence to show that the appropriate kernels are contained in the appropriate images. A choice for sections that work is simply multiplication composed with whatever number of identity maps are necessary. □

Proof of the theorem. So to prove the theorem we need to first consider the functor F : Mod_{A} → Mod_{A→B} which takes an A-module M to the pair (B ⊗_{A} M, ϕ_{M}) where ϕ_{M} : (B ⊗_{A} M) ⊗_{A} B → B ⊗_{A} (B ⊗_{A} M) maps b ⊗ m ⊗ b′ to b ⊗ b′ ⊗ m. To show that F is an equivalence of categories, we need to show that there is a functor G : Mod_{A→B} → Mod_{A} such that GF and F G are isomorphic to the identity. So let us define a functor G to take pairs (N, ϕ) to elements GN = {n ∈ N|1 ⊗ n = ϕ(n ⊗ 1)} and given a morphism β : (N, ϕ) → (N′, ϕ′) in Mod_{A→B} we get a morphism β_{G} : GN → GN′. So first let us check that GF is isomorphic to the identity. Notice that

((e_{1} − e_{2}) ⊗ id_{M})(b ⊗ m) = b ⊗ 1 ⊗ m − 1 ⊗ b ⊗ m = ϕ_{M}(b ⊗ m ⊗ 1) − 1 ⊗ b ⊗ m

for all m and b. For simplicity we can rewrite this as ((e_{1} − e_{2}) ⊗ id_{M})(x) = ϕ_{M}(x) − 1 ⊗ x for all x ∈ B ⊗_{A} M. So then by our definition of the functor G we get that G(B ⊗_{A} M, ϕ_{M}) = ker((e_{1} − e_{2}) ⊗ id_{M}). However, due to our lemma, we know that the sequence

0 → M →^{α_{M}} B ⊗_{A} M → B^{⊗2} ⊗_{A} M → B^{⊗3} ⊗_{A} M → ⋯

is exact, thus ker((e_{1} −e_{2})⊗id_{M}) = im(α_{M}) ≃ M. So M ≃ G(B ⊗_{A} M) = GF (M) as needed. Now we will show that F G is isomorphic to the identity. So we take (N, ϕ) in Mod_{A→B} and we set M = G(N, ϕ) = {n ∈ N|1 ⊗ n = ϕ(n ⊗ 1)}. Since M is an A-submodule of the B-submodule N we get a homomorphism of B-modules θ : B ⊗_{A} M → N which takes b ⊗ m to bm. It is easy to check that this is a morphism in Mod_{A→B}. So notice that we can also think of θ as a map F (M) → N, thus we can see that θ defines a natural transformation id → F G. So to complete the proof we need to show that θ is an isomorphism. First we will need to define some maps i will just be inclusion, ι_{M} : M ⊗B → B⊗M is the map taking m⊗b to b⊗m, and α, β : N → B⊗_{A}M are defined by α(n) = 1⊗n and β(n) = ϕ(n ⊗ 1). So by definition, M = ker(α − β). We have the following diagram, where the rows are exact:

0 ⟶ M ⊗_{A} B ⟶^{i⊗id_{B}} N ⊗_{A} B ⟶^{(α−β)⊗id_{B}} B ⊗_{A} N ⊗_{A} B
↓ θ ∘ ι_{M}  ↓ ϕ  ↓ ϕ_{1}
0 ⟶  N  ⟶^{α_{N}} B ⊗_{A} N ⟶^{(e_{1}−e_{2})⊗id_{N}} B ⊗_{A} B ⊗_{A} N

So by showing that the diagram commutes, using the fact that both ϕ and ϕ_{1} are isomorphisms we are able to get that θ is an isomorphism. To show the diagram commutes let us focus on one square at a time. For the first square we want ϕ(i ⊗ id_{B})(m ⊗ b) = α_{M}θι_{M}(m ⊗ b). We know that α_{M}θι_{M}(m ⊗ b) = 1 ⊗ bm. So we just need to show that we get the same thing for (ϕ(i ⊗ id_{B}))(m ⊗ b). We have

(ϕ(i ⊗ id_{B}))(b ⊗ m) = ϕ(m ⊗ b)
= ϕ((1 ⊗ b)(m ⊗ 1))
= (1 ⊗ b)ϕ(m ⊗ 1)
= (1 ⊗ b)(1 ⊗ m)
= 1 ⊗ bm

as needed. Now we just need to show the second square commutes. For the second square it should be clear that ϕ_{1}(α ⊗ id_{B}) = (e_{2} ⊗ id_{N}) ∘ ϕ. So we just need to check that ϕ_{1}(β ⊗ id_{B}) = (e_{1} ⊗ id_{N}) ∘ ϕ. We have:

ϕ_{1}(β ⊗ id_{B})(n ⊗ b) = ϕ_{1}(ϕ(n ⊗ 1) ⊗ b)
= ϕ_{1}ϕ_{3}(n ⊗ 1 ⊗ b)
= ϕ_{2}(n ⊗ 1 ⊗ b)
= (e_{1} ⊗ id_{N})ϕ(n ⊗ b)

So, by the argument above θ is an isomorphism and thus F G(N, ϕ) ≃ (N, ϕ). □

And now we can restate and sketch the proof of our desired result

Theorem 1.1.7. Let S be a scheme. The fibered category (QCoh/S) over (Sch/S) is stack with respect to the flat (fpqc) topology.

Proof. FIXME: Sketch of proof. Let me remind you that we just need to check the second condition of the lemma. So for a flat and surjective morphism V → U (corresponding to a faithfully flat ring homomorphism f : A → B). We need to show that there is an equivalence of categories between QCoh(U) and QCoh(V → U). We will do this using the previous theorem which states that there is an equivalence of categories between Mod_{A} and Mod_{A→B}. There is a standard equivalence of categories between QCoh(U) and Mod_{A}. So we just need to show that there is an equivalence of categories between QCoh(V → U) and Mod_{A→B}. To do this let us look at N an object in QCoh(V) which corresponds to an B-module N. Looking at p_{1}^{∗}N and p_{2}^{∗}N in V ×_{U} V = Spec(B ⊗_{A} B) we get N ⊗_{A} B and B ⊗_{A} N respectively. So the descent datum ψ : p_{1}^{∗}N ≃ p_{2}^{∗}N will correspond to the descent data ϕ : N ⊗_{A} B ≃ B ⊗_{A} N in Mod_{A→B}. So (N , ψ) is an object of QCoh(V → U) if and only if ϕ satisfies the cocycle condition, thus giving us an equivalence of categories between QCoh(V → U) and Mod_{A→B}. Thus the functor QCoh(U) → QCoh(V → U) corresponds to the functor Mod_{A} → Mod_{A→B}! So since, the later is an equivalence we get the equivalence of categories that we need. Thus finishing the proof. □

FIXME: put in example with descent of projective schemes and put in Galois descent example. To continue reading,

(1) visit the next section: The étale topology on schemes, Section 1, or

(2) go back to the table of contents: index.html#contents.

References

[Vis] Angelo Vistoli. Notes on grothendieck topologies, fibered categories and descent theory.