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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (2)
Comment #5518 by ykm on
Comment #5711 by Johan on