Let $A$ be a ring, $X$ a proper $A$ scheme, and $\mathscr{L}$ an ample invertible sheaf on $X$. Then we have that $X \cong \operatorname{Proj} \bigoplus_{n \ge 0} \Gamma(X, \mathscr{L}^{\otimes n})$. (see, for example, this question.)
Since $X$ is of finte type over $A$, it seems that $\bigoplus_{n \ge 0} \Gamma(X, \mathscr{L}^{\otimes n})$ (as a $\Gamma(X, \mathscr{O})$-algebra) is finitely generated. More precisely, if $\mathscr{L}^{\otimes N}$ is very ample, then it seems that $\bigoplus_{n \ge 0} \Gamma(X, \mathscr{L}^{\otimes n})$ is generated by $\bigoplus_{N \gt n \ge 0} \Gamma(X, \mathscr{L}^{\otimes n})$.
How can I show it?
