Sections of sheaves of modules: reference

Solution 1:

Saying that a sheaf $\mathcal{F}$ is generated by global sections is the same as saying that there is an exact seqeunce

$$\bigoplus_{i\in I}\mathcal{O}_X\to\mathcal{F}\to0.$$

Lemma 17.4.2 is basically the fact that this sequence is exact if and only if it is exact at the level of stalks, while Lemma 17.4.3 corresponds to the fact that the functor $-\otimes_{\mathcal{O}_X}\mathcal{G}$ is right-exact, so you can apply it to the above sequence and get a sequence

$$\bigoplus_{i\in I}\mathcal{O}_X\otimes_{\mathcal{O}_X}\mathcal{G}\to\mathcal{F}\otimes_{\mathcal{O}_X}\mathcal{G}\to0,$$

which is $$\bigoplus_{i\in I}\mathcal{G}\to\mathcal{F}\otimes_{\mathcal{O}_X}\mathcal{G}\to0,$$

and then you can surject upon each term on the left using a different direct sum of structure sheaves.