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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 06VK. Beware of the difference between the letter 'O' and the digit '0'.