Proposition 59.17.1 (03OG)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.17: Quasi-coherent sheaves
Proposition 59.17.1 (cite)

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Proposition 59.17.1. For any quasi-coherent sheaf $\mathcal{F}$ on $S$ the presheaf

$\begin{matrix} \mathcal{F}^ a : & \mathit{Sch}/S & \to & \textit{Ab} \\ & (f: T \to S) & \mapsto & \Gamma (T, f^*\mathcal{F}) \end{matrix}$

is an $\mathcal{O}$ -module which satisfies the sheaf condition for the fpqc topology.

Proof. This is proved in Descent, Lemma 35.8.1. We indicate the proof here. As established in Lemma 59.15.6, it is enough to check the sheaf property on Zariski coverings and faithfully flat morphisms of affine schemes. The sheaf property for Zariski coverings is standard scheme theory, since $\Gamma (U, i^\ast \mathcal{F}) = \mathcal{F}(U)$ when $i : U \hookrightarrow S$ is an open immersion.

For $\left\{ \mathop{\mathrm{Spec}}(B)\to \mathop{\mathrm{Spec}}(A)\right\}$ with $A\to B$ faithfully flat and $\mathcal{F}|_{\mathop{\mathrm{Spec}}(A)} = \widetilde{M}$ this corresponds to the fact that $M = H^0\left((B/A)_\bullet \otimes _ A M \right)$ , i.e., that

$\begin{align*} 0 \to M \to B \otimes _ A M \to B \otimes _ A B \otimes _ A M \end{align*}$

is exact by Lemma 59.16.4. $\square$

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