Lemma 95.4.2. The functor $p : \mathcal{QC}\! \mathit{oh}\to (\mathit{Sch}/S)_{fppf}$ satisfies conditions (1), (2) and (3) of Stacks, Definition 8.4.1.

**Proof.**
It is clear from Lemma 95.4.1 that $\mathcal{QC}\! \mathit{oh}$ is a fibred category over $(\mathit{Sch}/S)_{fppf}$. Given covering $\mathcal{U} = \{ X_ i \to X\} _{i \in I}$ of $(\mathit{Sch}/S)_{fppf}$ the functor

is fully faithful and essentially surjective, see Descent, Proposition 35.5.2. Hence Stacks, Lemma 8.4.2 applies to show that $\mathcal{QC}\! \mathit{oh}$ satisfies all the axioms of a stack. $\square$

## 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Your email address will not be published.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.

## Comments (2)

Comment #5518 by ykm on

Comment #5711 by Johan on