The Stacks project

Lemma 46.5.10. Let $S$ be a scheme. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules on $(\mathit{Sch}/S)_\tau $. If for every affine scheme $\mathop{\mathrm{Spec}}(A)$ over $S$ the functor $F_{\mathcal{F}, A}$ is adequate, then the sheafification of $\mathcal{F}$ is an adequate $\mathcal{O}$-module.

Proof. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme over $S$. Set $F = F_{\mathcal{F}, A}$. The sheafification $\mathcal{F}^\# = (\mathcal{F}^+)^+$, see Sites, Section 7.10. By construction

\[ (\mathcal{F})^+(U) = \mathop{\mathrm{colim}}\nolimits _\mathcal {U} \check{H}^0(\mathcal{U}, \mathcal{F}) \]

where the colimit is over coverings in the site $(\mathit{Sch}/S)_\tau $. Since $U$ is affine it suffices to take the limit over standard affine $\tau $-coverings $\mathcal{U} = \{ U_ i \to U\} _{i \in I} = \{ \mathop{\mathrm{Spec}}(A_ i) \to \mathop{\mathrm{Spec}}(A)\} _{i \in I}$ of $U$. Since each $A \to A_ i$ and $A \to A_ i \otimes _ A A_ j$ is flat we see that

\[ \check{H}^0(\mathcal{U}, \mathcal{F}) = \mathop{\mathrm{Ker}}(\prod F(A) \otimes _ A A_ i \to \prod F(A) \otimes _ A A_ i \otimes _ A A_ j) \]

by Lemma 46.3.5. Since $A \to \prod A_ i$ is faithfully flat we see that this always is canonically isomorphic to $F(A)$ by Descent, Lemma 35.3.6. Thus the presheaf $(\mathcal{F})^+$ has the same value as $\mathcal{F}$ on all affine schemes over $S$. Repeating the argument once more we deduce the same thing for $\mathcal{F}^\# = ((\mathcal{F})^+)^+$. Thus $F_{\mathcal{F}, A} = F_{\mathcal{F}^\# , A}$ and we conclude that $\mathcal{F}^\# $ is adequate. $\square$

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