Proposition 58.17.1. For any quasi-coherent sheaf $\mathcal{F}$ on $S$ the presheaf

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

Proposition 58.17.1. 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 58.15.6, 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 58.16.4. $\square$

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