The Stacks project

Lemma 94.4.2. 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 94.4.1 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

\[ \mathit{QCoh}(\mathcal{O}_ X) \longrightarrow DD(\mathcal{U}) \]

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. $\square$

Comments (2)

Comment #5518 by ykm on

QCoh(O_T)\to DD(U) i think the T should be X

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