Lemma 46.5.5. Let $S$ be a scheme. Let $\mathcal{F}$ be an $\mathcal{O}$-module on $(\mathit{Sch}/S)_\tau $. The following are equivalent

$\mathcal{F}$ is adequate,

there exists an affine open covering $S = \bigcup S_ i$ and maps of quasi-coherent $\mathcal{O}_{S_ i}$-modules $\mathcal{G}_ i \to \mathcal{H}_ i$ such that $\mathcal{F}|_{(\mathit{Sch}/S_ i)_\tau }$ is the kernel of $\mathcal{G}_ i^ a \to \mathcal{H}_ i^ a$

there exists a $\tau $-covering $\{ S_ i \to S\} _{i \in I}$ and maps of $\mathcal{O}_{S_ i}$-quasi-coherent modules $\mathcal{G}_ i \to \mathcal{H}_ i$ such that $\mathcal{F}|_{(\mathit{Sch}/S_ i)_\tau }$ is the kernel of $\mathcal{G}_ i^ a \to \mathcal{H}_ i^ a$,

there exists a $\tau $-covering $\{ f_ i : S_ i \to S\} _{i \in I}$ such that each $f_ i^*\mathcal{F}$ is adequate,

for any affine scheme $U$ over $S$ the restriction $\mathcal{F}|_{(\mathit{Sch}/U)_\tau }$ is the kernel of a map $\mathcal{G}^ a \to \mathcal{H}^ a$ of quasi-coherent $\mathcal{O}_ U$-modules.

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