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35.5 Fpqc descent of quasi-coherent sheaves

The main application of flat descent for modules is the corresponding descent statement for quasi-coherent sheaves with respect to fpqc-coverings.

Lemma 35.5.1. Let $S$ be an affine scheme. Let $\mathcal{U} = \{ f_ i : U_ i \to S\} _{i = 1, \ldots , n}$ be a standard fpqc covering of $S$, see Topologies, Definition 34.9.9. Any descent datum on quasi-coherent sheaves for $\mathcal{U} = \{ U_ i \to S\} $ is effective. Moreover, the functor from the category of quasi-coherent $\mathcal{O}_ S$-modules to the category of descent data with respect to $\mathcal{U}$ is fully faithful.

Proof. This is a restatement of Proposition 35.3.9 in terms of schemes. First, note that a descent datum $\xi $ for quasi-coherent sheaves with respect to $\mathcal{U}$ is exactly the same as a descent datum $\xi '$ for quasi-coherent sheaves with respect to the covering $\mathcal{U}' = \{ \coprod _{i = 1, \ldots , n} U_ i \to S\} $. Moreover, effectivity for $\xi $ is the same as effectivity for $\xi '$. Hence we may assume $n = 1$, i.e., $\mathcal{U} = \{ U \to S\} $ where $U$ and $S$ are affine. In this case descent data correspond to descent data on modules with respect to the ring map

\[ \Gamma (S, \mathcal{O}) \longrightarrow \Gamma (U, \mathcal{O}). \]

Since $U \to S$ is surjective and flat, we see that this ring map is faithfully flat. In other words, Proposition 35.3.9 applies and we win. $\square$

Proposition 35.5.2. Let $S$ be a scheme. Let $\mathcal{U} = \{ \varphi _ i : U_ i \to S\} $ be an fpqc covering, see Topologies, Definition 34.9.1. Any descent datum on quasi-coherent sheaves for $\mathcal{U} = \{ U_ i \to S\} $ is effective. Moreover, the functor from the category of quasi-coherent $\mathcal{O}_ S$-modules to the category of descent data with respect to $\mathcal{U}$ is fully faithful.

Proof. Let $S = \bigcup _{j \in J} V_ j$ be an affine open covering. For $j, j' \in J$ we denote $V_{jj'} = V_ j \cap V_{j'}$ the intersection (which need not be affine). For $V \subset S$ open we denote $\mathcal{U}_ V = \{ V \times _ S U_ i \to V\} _{i \in I}$ which is a fpqc-covering (Topologies, Lemma 34.9.7). By definition of an fpqc covering, we can find for each $j \in J$ a finite set $K_ j$, a map $\underline{i} : K_ j \to I$, affine opens $U_{\underline{i}(k), k} \subset U_{\underline{i}(k)}$, $k \in K_ j$ such that $\mathcal{V}_ j = \{ U_{\underline{i}(k), k} \to V_ j\} _{k \in K_ j}$ is a standard fpqc covering of $V_ j$. And of course, $\mathcal{V}_ j$ is a refinement of $\mathcal{U}_{V_ j}$. Picture

\[ \xymatrix{ \mathcal{V}_ j \ar[r] \ar@{~>}[d] & \mathcal{U}_{V_ j} \ar[r] \ar@{~>}[d] & \mathcal{U} \ar@{~>}[d] \\ V_ j \ar@{=}[r] & V_ j \ar[r] & S } \]

where the top horizontal arrows are morphisms of families of morphisms with fixed target (see Sites, Definition 7.8.1).

To prove the proposition you show successively the faithfulness, fullness, and essential surjectivity of the functor from quasi-coherent sheaves to descent data.

Faithfulness. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent sheaves on $S$ and let $a, b : \mathcal{F} \to \mathcal{G}$ be homomorphisms of $\mathcal{O}_ S$-modules. Suppose $\varphi _ i^*(a) = \varphi ^*(b)$ for all $i$. Pick $s \in S$. Then $s = \varphi _ i(u)$ for some $i \in I$ and $u \in U_ i$. Since $\mathcal{O}_{S, s} \to \mathcal{O}_{U_ i, u}$ is flat, hence faithfully flat (Algebra, Lemma 10.38.17) we see that $a_ s = b_ s : \mathcal{F}_ s \to \mathcal{G}_ s$. Hence $a = b$.

Fully faithfulness. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent sheaves on $S$ and let $a_ i : \varphi _ i^*\mathcal{F} \to \varphi _ i^*\mathcal{G}$ be homomorphisms of $\mathcal{O}_{U_ i}$-modules such that $\text{pr}_0^*a_ i = \text{pr}_1^*a_ j$ on $U_ i \times _ U U_ j$. We can pull back these morphisms to get morphisms

\[ a_ k : \mathcal{F}|_{U_{\underline{i}(k), k}} \longrightarrow \mathcal{G}|_{U_{\underline{i}(k), k}} \]

$k \in K_ j$ with notation as above. Moreover, Lemma 35.2.2 assures us that these define a morphism between (canonical) descent data on $\mathcal{V}_ j$. Hence, by Lemma 35.5.1, we get correspondingly unique morphisms $a_ j : \mathcal{F}|_{V_ j} \to \mathcal{G}|_{V_ j}$. To see that $a_ j|_{V_{jj'}} = a_{j'}|_{V_{jj'}}$ we use that both $a_ j$ and $a_{j'}$ agree with the pullback of the morphism $(a_ i)_{i \in I}$ of (canonical) descent data to any covering refining both $\mathcal{V}_{j, V_{jj'}}$ and $\mathcal{V}_{j', V_{jj'}}$, and using the faithfulness already shown. For example the covering $\mathcal{V}_{jj'} = \{ V_ k \times _ S V_{k'} \to V_{jj'}\} _{k \in K_ j, k' \in K_{j'}}$ will do.

Essential surjectivity. Let $\xi = (\mathcal{F}_ i, \varphi _{ii'})$ be a descent datum for quasi-coherent sheaves relative to the covering $\mathcal{U}$. Pull back this descent datum to get descent data $\xi _ j$ for quasi-coherent sheaves relative to the coverings $\mathcal{V}_ j$ of $V_ j$. By Lemma 35.5.1 once again there exist quasi-coherent sheaves $\mathcal{F}_ j$ on $V_ j$ whose associated canonical descent datum is isomorphic to $\xi _ j$. By fully faithfulness (proved above) we see there are isomorphisms

\[ \phi _{jj'} : \mathcal{F}_ j|_{V_{jj'}} \longrightarrow \mathcal{F}_{j'}|_{V_{jj'}} \]

corresponding to the isomorphism of descent data between the pullback of $\xi _ j$ and $\xi _{j'}$ to $\mathcal{V}_{jj'}$. To see that these maps $\phi _{jj'}$ satisfy the cocycle condition we use faithfulness (proved above) over the triple intersections $V_{jj'j''}$. Hence, by Lemma 35.2.4 we see that the sheaves $\mathcal{F}_ j$ glue to a quasi-coherent sheaf $\mathcal{F}$ as desired. We still have to verify that the canonical descent datum relative to $\mathcal{U}$ associated to $\mathcal{F}$ is isomorphic to the descent datum we started out with. This verification is omitted. $\square$


Comments (4)

Comment #1534 by jojo on

I get a parsing error for the first equation of the proof.

Comment #2962 by Zhon Lin on

It seems that the statement of prop 34.5.2 can be improved (the statement only says the functor is fully faithful while the proof also shows it is essentially surjective).

Comment #3088 by on

@#2962 OK, but the statement also says every descent datum is effective which is the essential surjectivity.


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