17.4 Sections of sheaves of modules
Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. Let $s \in \Gamma (X, \mathcal{F}) = \mathcal{F}(X)$ be a global section. There is a unique map of $\mathcal{O}_ X$-modules
\[ \mathcal{O}_ X \longrightarrow \mathcal{F}, \ f \longmapsto fs \]
associated to $s$. The notation above signifies that a local section $f$ of $\mathcal{O}_ X$, i.e., a section $f$ over some open $U$, is mapped to the multiplication of $f$ with the restriction of $s$ to $U$. Conversely, any map $\varphi : \mathcal{O}_ X \to \mathcal{F}$ gives rise to a section $s = \varphi (1)$ such that $\varphi $ is the morphism associated to $s$.
Definition 17.4.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. We say that $\mathcal{F}$ is generated by global sections if there exist a set $I$, and global sections $s_ i \in \Gamma (X, \mathcal{F})$, $i \in I$ such that the map
\[ \bigoplus \nolimits _{i \in I} \mathcal{O}_ X \longrightarrow \mathcal{F} \]
which is the map associated to $s_ i$ on the summand corresponding to $i$, is surjective. In this case we say that the sections $s_ i$ generate $\mathcal{F}$.
We often use the abuse of notation introduced in Sheaves, Section 6.11 where, given a local section $s$ of $\mathcal{F}$ defined in an open neighbourhood of a point $x \in X$, we denote $s_ x$, or even $s$ the image of $s$ in the stalk $\mathcal{F}_ x$.
Lemma 17.4.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. Let $I$ be a set. Let $s_ i \in \Gamma (X, \mathcal{F})$, $i \in I$ be global sections. The sections $s_ i$ generate $\mathcal{F}$ if and only if for all $x\in X$ the elements $s_{i, x} \in \mathcal{F}_ x$ generate the $\mathcal{O}_{X, x}$-module $\mathcal{F}_ x$.
Proof.
Omitted.
$\square$
slogan
Lemma 17.4.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}_ X$-modules. If $\mathcal{F}$ and $\mathcal{G}$ are generated by global sections then so is $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$.
Proof.
Omitted.
$\square$
Lemma 17.4.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. Let $I$ be a set. Let $s_ i$, $i \in I$ be a collection of local sections of $\mathcal{F}$, i.e., $s_ i \in \mathcal{F}(U_ i)$ for some opens $U_ i \subset X$. There exists a unique smallest subsheaf of $\mathcal{O}_ X$-modules $\mathcal{G}$ such that each $s_ i$ corresponds to a local section of $\mathcal{G}$.
Proof.
Consider the subpresheaf of $\mathcal{O}_ X$-modules defined by the rule
\[ U \longmapsto \{ \text{sums } \sum \nolimits _{i \in J} f_ i (s_ i|_ U) \text{ where } J \text{ is finite, } U \subset U_ i \text{ for } i\in J, \text{ and } f_ i \in \mathcal{O}_ X(U) \} \]
Let $\mathcal{G}$ be the sheafification of this subpresheaf. This is a subsheaf of $\mathcal{F}$ by Sheaves, Lemma 6.16.3. Since all the finite sums clearly have to be in $\mathcal{G}$ this is the smallest subsheaf as desired.
$\square$
Definition 17.4.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. Given a set $I$, and local sections $s_ i$, $i \in I$ of $\mathcal{F}$ we say that the subsheaf $\mathcal{G}$ of Lemma 17.4.4 above is the subsheaf generated by the $s_ i$.
Lemma 17.4.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. Given a set $I$, and local sections $s_ i$, $i \in I$ of $\mathcal{F}$. Let $\mathcal{G}$ be the subsheaf generated by the $s_ i$ and let $x\in X$. Then $\mathcal{G}_ x$ is the $\mathcal{O}_{X, x}$-submodule of $\mathcal{F}_ x$ generated by the elements $s_{i, x}$ for those $i$ such that $s_ i$ is defined at $x$.
Proof.
This is clear from the construction of $\mathcal{G}$ in the proof of Lemma 17.4.4.
$\square$
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