The Stacks project

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

  1. $\mathcal{F}$ is adequate,

  2. 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 $\mathcal{G}_ i^ a \to \mathcal{H}_ i^ a$

  3. 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 $\mathcal{G}_ i^ a \to \mathcal{H}_ i^ a$,

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

  5. 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.

Proof. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme over $S$. Set $F = F_{\mathcal{F}, A}$. By definition, the functor $F$ is adequate if and only if there exists a map of $A$-modules $M \to N$ such that $F = \mathop{\mathrm{Ker}}(\underline{M} \to \underline{N})$. Combining with Lemmas 46.5.2 and 46.5.3 we see that (1) and (5) are equivalent.

It is clear that (5) implies (2) and (2) implies (3). If (3) holds then we can refine the covering $\{ S_ i \to S\} $ such that each $S_ i = \mathop{\mathrm{Spec}}(A_ i)$ is affine. Then we see, by the preliminary remarks of the proof, that $F_{\mathcal{F}, A_ i}$ is adequate. Thus $\mathcal{F}$ is adequate by definition. Hence (3) implies (1).

Finally, (4) is equivalent to (1) using Lemma 46.5.4 for one direction and that a composition of $\tau $-coverings is a $\tau $-covering for the other. $\square$

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