The Stacks project

Proposition 72.4.1. Let $S$ be a scheme. Let $\{ X_ i \to X\} $ be an fpqc covering of algebraic spaces over $S$, see Topologies on Spaces, Definition 71.9.1. Any descent datum on quasi-coherent sheaves for $\{ X_ i \to X\} $ is effective. Moreover, the functor from the category of quasi-coherent $\mathcal{O}_ X$-modules to the category of descent data with respect to $\{ X_ i \to X\} $ is fully faithful.

Proof. This is more or less a formal consequence of the corresponding result for schemes, see Descent, Proposition 35.5.2. Here is a strategy for a proof:

  1. The fact that $\{ X_ i \to X\} $ is a refinement of the trivial covering $\{ X \to X\} $ gives, via Lemma 72.3.2, a functor $\mathit{QCoh}(\mathcal{O}_ X) \to DD(\{ X_ i \to X\} )$ from the category of quasi-coherent $\mathcal{O}_ X$-modules to the category of descent data for the given family.

  2. In order to prove the proposition we will construct a quasi-inverse functor $back : DD(\{ X_ i \to X\} ) \to \mathit{QCoh}(\mathcal{O}_ X)$.

  3. Applying again Lemma 72.3.2 we see that there is a functor $DD(\{ X_ i \to X\} ) \to DD(\{ T_ j \to X\} )$ if $\{ T_ j \to X\} $ is a refinement of the given family. Hence in order to construct the functor $back$ we may assume that each $X_ i$ is a scheme, see Topologies on Spaces, Lemma 71.9.5. This reduces us to the case where all the $X_ i$ are schemes.

  4. A quasi-coherent sheaf on $X$ is by definition a quasi-coherent $\mathcal{O}_ X$-module on $X_{\acute{e}tale}$. Now for any $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ we get an fppf covering $\{ U_ i \times _ X X_ i \to U\} $ by schemes and a morphism $g : \{ U_ i \times _ X X_ i \to U\} \to \{ X_ i \to X\} $ of coverings lying over $U \to X$. Given a descent datum $\xi = (\mathcal{F}_ i, \varphi _{ij})$ we obtain a quasi-coherent $\mathcal{O}_ U$-module $\mathcal{F}_{\xi , U}$ corresponding to the pullback $g^*\xi $ of Lemma 72.3.2 to the covering of $U$ and using effectivity for fppf covering of schemes, see Descent, Proposition 35.5.2.

  5. Check that $\xi \mapsto \mathcal{F}_{\xi , U}$ is functorial in $\xi $. Omitted.

  6. Check that $\xi \mapsto \mathcal{F}_{\xi , U}$ is compatible with morphisms $U \to U'$ of the site $X_{\acute{e}tale}$, so that the system of sheaves $\mathcal{F}_{\xi , U}$ corresponds to a quasi-coherent $\mathcal{F}_\xi $ on $X_{\acute{e}tale}$, see Properties of Spaces, Lemma 64.29.3. Details omitted.

  7. Check that $back : \xi \mapsto \mathcal{F}_\xi $ is quasi-inverse to the functor constructed in (1). Omitted.

This finishes the proof. $\square$

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