I am wondering why the $\tilde{M}$ is defined like that.
To be more precise, when $S=\mathbb{C}[x_0,...,x_n]$. $X:=$Proj $S=\mathbb{P}_\mathbb{C}^n$. $\tilde{M}$ is similar to the sheaf defined on the projective space $\mathbb{C}P^n$. In particular, $\mathcal{O}_X(k)$ is like the line bundle on $\mathbb{C}P^n$. My question is: Is there a motivation that $\mathcal{O}_X(k)$ is defined to be the set of functions $s$ from $U$ to $\coprod_{p\in U}S(k)_{(p)}$? How is this definition related to the definition of line bundle or sheaf on $\mathbb{C}P^n$? What's the motivation of using the localization here for $S(k)_{(p)}$?