The Stacks project

Proposition 56.17.1. 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 34.8.1. We indicate the proof here. As established in Lemma 56.15.6, 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 56.16.4. $\square$


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