The Stacks project

Proposition 74.4.1. Let $S$ be a scheme. Let $\{ X_ i \to X\} $ be an fpqc covering of algebraic spaces over $S$, see Topologies on Spaces, Definition 73.9.1. 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 74.3.2, 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 74.3.2 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 73.9.5. 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 74.3.2 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 66.29.3. 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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04W8. Beware of the difference between the letter 'O' and the digit '0'.