Clarification on Ravi Vakil's AG notes, Exercise 13.3.H

Question

Let me first write what this exercise is:

[...] Suppose $X$ is a quasicompact quasiseparated scheme, $\mathscr{L}$ is an invertible sheaf on $X$ with section $s$, and $\mathscr{F}$ a quasicoherent sheaf on $X$. [...] Let $X_s$ be the open subset of $X$ where $s$ doesn't vanish. [...] Note that $\bigoplus_{n\geq 0}\Gamma(X, \mathscr{L}^{\otimes n})$ is a graded ring, and we interpret $s$ as a degree one. Note also that $\bigoplus_{n\geq 0} \Gamma(X,\mathscr{F}\otimes_{\mathscr{O}_X}\mathscr{L}^{\otimes n})$ is a graded module over this ring. Describe a natural map $$\left[\left(\bigoplus_{n\geq 0}\Gamma(X,\mathscr{F}\otimes_{\mathscr{O}_X}\mathscr{L}^{\otimes n})\right)_s\:\right]_0\to \Gamma(X_s, \mathscr{F})$$ and show that it is an isomorphism.

I'm going to list the things I'm confused about (mostly trying to understand what this question means) and what I think might be the answer to each:

1. "Let $X_s$ be the open subset of $X$ where $s$ doesn't vanish." The only way I can make sense of $X_s$ is as follows: There is a an affine open covering of $X$, $U_i$ with a sheaf isomorphism $\phi_i:\mathscr{L}|_{U_i}\simeq \mathscr{O}_{U_i}$. Define $\sigma_i:=\phi_i(U_i)(s|_{U_i})$. Vanishing or non-vanishing of $\sigma_i$ is meaningful so we define $X_s=\bigcup_{i\in I} D_i(\sigma_i)$ where $D_i(f)\subset U_i:=\mathrm{Spec} R_i$ is the distinguished open subset of affine scheme $U_i$ coming from $f\in R_i$. Is this indeed what $X_s$ means?
2. The second issue is the grading structure on $\bigoplus_{n\geq 0} \Gamma(X,\mathscr{L}^{\otimes n})$ and $\bigoplus_{n\geq 0} \Gamma(X,\mathscr{F}\otimes\mathscr{L}^{\otimes n})$. The only kind of of multiplication I can think of is the tensor product. Thankfully if $\mathscr{G}, \mathscr{G}'$ are two $\mathscr{O}_X$-modules, then the sheafification morphism $$ \eta: [\mathscr{G}\otimes \mathscr{G}']_\text{pre}\to \mathscr{G}\otimes \mathscr{G}' $$ induces a natural map $\Gamma(X,\mathscr{G})\otimes_{\Gamma(X,\mathscr{O}_X)}\Gamma(X, \mathscr{G}')\to \Gamma(X, \mathscr{G}\otimes \mathscr{G}')$. So in our case we have maps $$ \begin{aligned} &\Gamma(X,\mathscr{L}^{\otimes n})\otimes_{\Gamma(X,\mathscr{O}_X)}\Gamma(X, \mathscr{L}^{\otimes m})\to \Gamma(X, \mathscr{L}^{\otimes (n+m)})\\ &\Gamma(X,\mathscr{L}^{\otimes n})\otimes_{\Gamma(X,\mathscr{O}_X)}\Gamma(X, \mathscr{F}\otimes \mathscr{L}^{\otimes m})\to \Gamma(X, \mathscr{F}\otimes\mathscr{L}^{\otimes (n+m)}) \end{aligned} $$ This looks promising, for a good grading structure... But one problem: Are these maps necessarily injective/inclusion?

Of course I haven't even begun yet to actually solve this problem. But this question has already become too long, so I'll leave it at that.

Answer 1

$1)$ Your definition of $X_s$ is correct, you can see exercise $13.1.I.$ for the definition of the vanishing scheme of a section of a loccaly free sheaf and then take the complement. You can also use the more canonical point of view: for any sheaf of $\mathcal{O}_X$ modules, $\mathscr{F}$, with global section $s$, we say $s$ vanishes at $p$ if $s_p \in \mathfrak{m}_{X,p}\cdot\mathscr{F}_p$ and we can show that the collection of all such points is a closed subset. (Although without the further assumption that $\mathscr{F}$ is locally free, it isn't naturally a closed subscheme.)

$2)$ This is indeed the requisite map, and there's no need for it to be an inclusion (for a graded module $M_{\bullet}$, there's no requirement that the maps $M_n \times M_m \rightarrow M_{n+m}$ are an inclusion).