The Stacks project

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.