Sections of sheaves of modules: reference

Question

I try to understand Section 17.4 in the Stack Project talking about sections of sheaves of modules and their stalks. Especially, I would like to study Lemma 17.4.2 and 17.4.3 there, but the proofs are omitted there.

Does anyone know a good reference like Hartshorne, Shafaverich, etc?

Answer 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.