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
Lemma 17.4.3.slogan 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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