## 18.17 Global types of modules

Definition 18.17.1. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules.

1. We say $\mathcal{F}$ is a free $\mathcal{O}$-module if $\mathcal{F}$ is isomorphic as an $\mathcal{O}$-module to a sheaf of the form $\bigoplus _{i \in I} \mathcal{O}$.

2. We say $\mathcal{F}$ is finite free if $\mathcal{F}$ is isomorphic as an $\mathcal{O}$-module to a sheaf of the form $\bigoplus _{i \in I} \mathcal{O}$ with a finite index set $I$.

3. We say $\mathcal{F}$ is generated by global sections if there exists a surjection

$\bigoplus \nolimits _{i \in I} \mathcal{O} \longrightarrow \mathcal{F}$

from a free $\mathcal{O}$-module onto $\mathcal{F}$.

4. Given $r \geq 0$ we say $\mathcal{F}$ is generated by $r$ global sections if there exists a surjection $\mathcal{O}^{\oplus r} \to \mathcal{F}$.

5. We say $\mathcal{F}$ is generated by finitely many global sections if it is generated by $r$ global sections for some $r \geq 0$.

6. We say $\mathcal{F}$ has a global presentation if there exists an exact sequence

$\bigoplus \nolimits _{j \in J} \mathcal{O} \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O} \longrightarrow \mathcal{F} \longrightarrow 0$

of $\mathcal{O}$-modules.

7. We say $\mathcal{F}$ has a global finite presentation if there exists an exact sequence

$\bigoplus \nolimits _{j \in J} \mathcal{O} \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O} \longrightarrow \mathcal{F} \longrightarrow 0$

of $\mathcal{O}$-modules with $I$ and $J$ finite sets.

Note that for any set $I$ the direct sum $\bigoplus _{i \in I} \mathcal{O}$ exists (Lemma 18.14.2) and is the sheafification of the presheaf $U \mapsto \bigoplus _{i \in I} \mathcal{O}(U)$. This module is called the free $\mathcal{O}$-module on the set $I$.

Lemma 18.17.2. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {D}$-module.

1. If $\mathcal{F}$ is free then $f^*\mathcal{F}$ is free.

2. If $\mathcal{F}$ is finite free then $f^*\mathcal{F}$ is finite free.

3. If $\mathcal{F}$ is generated by global sections then $f^*\mathcal{F}$ is generated by global sections.

4. Given $r \geq 0$ if $\mathcal{F}$ is generated by $r$ global sections, then $f^*\mathcal{F}$ is generated by $r$ global sections.

5. If $\mathcal{F}$ is generated by finitely many global sections then $f^*\mathcal{F}$ is generated by finitely many global sections.

6. If $\mathcal{F}$ has a global presentation then $f^*\mathcal{F}$ has a global presentation.

7. If $\mathcal{F}$ has a finite global presentation then $f^*\mathcal{F}$ has a finite global presentation.

Proof. This is true because $f^*$ commutes with arbitrary colimits (Lemma 18.14.3) and $f^*\mathcal{O}_\mathcal {D} = \mathcal{O}_\mathcal {C}$. $\square$

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