The Stacks project

66.30 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 $\mathcal{P}$:

1. free,

2. finite free,

3. generated by global sections,

4. generated by finitely many global sections,

5. having a global presentation, and

6. having a global finite presentation.

Local properties $\mathcal{P}$:

1. locally free,

2. finite locally free,

3. locally generated by sections,

4. locally generated by $r$ sections,

5. finite type,

6. quasi-coherent (see Section 66.29),

7. of finite presentation,

8. coherent, and

9. flat.

Here are some results which follow immediately from the definitions:

1. 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.

2. 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 66.18.11).

3. 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.

4. 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.

5. 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.