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.

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

Given $r \geq 0$ we say $\mathcal{F}$ is

*generated by $r$ global sections*if there exists a surjection $\mathcal{O}^{\oplus r} \to \mathcal{F}$.We say $\mathcal{F}$ is

*generated by finitely many global sections*if it is generated by $r$ global sections for some $r \geq 0$.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.

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.

## Comments (2)

Comment #1151 by Olaf SchnÃ¼rer on

Comment #1172 by Johan on