## 65.30 Properties of modules

In Modules on Sites, Sections 18.17, 18.23, and Definition 18.28.1 we have defined a number of intrinsic properties of modules of $\mathcal{O}$-module on any ringed topos. If $X$ is an algebraic space, we will apply these notions freely to modules on the ringed site $(X_{\acute{e}tale}, \mathcal{O}_ X)$, or equivalently on the ringed site $(X_{spaces, {\acute{e}tale}}, \mathcal{O}_ X)$.

Global properties $\mathcal{P}$:

*free*,*finite free*,*generated by global sections*,*generated by finitely many global sections*,having a

*global presentation*, andhaving a

*global finite presentation*.

Local properties $\mathcal{P}$:

*locally free*,*finite locally free*,*locally generated by sections*,*locally generated by $r$ sections*,*finite type*,*quasi-coherent*(see Section 65.29),*of finite presentation*,*coherent*, and*flat*.

Here are some results which follow immediately from the definitions:

In each case, except for $\mathcal{P}=$“coherent”, the property is preserved under pullback, see Modules on Sites, Lemmas 18.17.2, 18.23.4, and 18.39.1.

Each of the properties above (including coherent) are preserved under pullbacks by étale morphisms of algebraic spaces (because in this case pullback is given by restriction, see Lemma 65.18.11).

Assume $f : Y \to X$ is a surjective étale morphism of algebraic spaces. For each of the local properties (g) – (m), the fact that $f^*\mathcal{F}$ has $\mathcal{P}$ implies that $\mathcal{F}$ has $\mathcal{P}$. This follows as $\{ Y \to X\} $ is a covering in $X_{spaces, {\acute{e}tale}}$ and Modules on Sites, Lemma 18.23.3.

If $X$ is a scheme, $\mathcal{F}$ is a quasi-coherent module on $X_{\acute{e}tale}$, and $\mathcal{P}$ any property except “coherent” or “locally free”, then $\mathcal{P}$ for $\mathcal{F}$ on $X_{\acute{e}tale}$ is equivalent to the corresponding property for $\mathcal{F}|_{X_{Zar}}$, i.e., it corresponds to $\mathcal{P}$ for $\mathcal{F}$ when we think of it as a quasi-coherent sheaf on the scheme $X$. See Descent, Lemma 35.8.10.

If $X$ is a locally Noetherian scheme, $\mathcal{F}$ is a quasi-coherent module on $X_{\acute{e}tale}$, then $\mathcal{F}$ is coherent on $X_{\acute{e}tale}$ if and only if $\mathcal{F}|_{X_{Zar}}$ is coherent, i.e., it corresponds to the usual notion of a coherent sheaf on the scheme $X$ being coherent. See Descent, Lemma 35.8.10.

## 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Your email address will not be published.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.

## Comments (0)