TITLE: Finite generated sheaf of graded $\mathcal{O}_X$ algebra
QUESTION [3 upvotes]: Let $X$ be a projective normal variety. 
I am having trouble finding out what it means for a graded sheaf of $\mathcal{O}_X$ algebra $R$ to be finitely generated over $\mathcal{O}_X$. Does it mean that for each affine open $U=\operatorname{Spec} A\subset X$, one has $R|_U$ is a finitely generated graded algebra over $A$?
Also how does one see that if $f:X\to Y$ is a binational morphism with $K_X$ $f$-ample, then $R=\bigoplus_{n\geq 0}f_*(\mathcal{O}_X(nK_X))$ is a finitely generated $\mathcal{O}_Y$-algebra?
Thanks for the help!

REPLY [3 votes]: Yes a sheaf $R$ of graded $\mathcal{O}_X$-algebras is finitely generated, if for every open affine $U$, $R(U)$ is a finitely generated $\mathcal{O}_X(U)$-algebra.
To answer your second question, notice first a basic result about graded rings. If $R$ is a graded $A$-algebra then $R$ is finitely generated if and only if $R_{(d)}$ is finitely generated for some positive integer $d$, where$$R_{(d)} = \bigoplus_m R_{md},$$the graded pieces of $R$ of degree divisible by $d$ (it is also usual to rescale the grading so that $R_d$ has degree one in $R_{(d)}$, etc). This is basically due to Noether.
So now suppose that $f: X \to Y$ is projective and $D$ is any integral $\mathbb{Q}$-Cartier divisor which is relatively ample. Then the sheaf $R$ of graded algebras is finitely generated. By the observation above we are free to replace $D$ with a multiple. So we may assume that $D = \mathcal{O}_X(1)$, for some embedding of $X$ in $\mathbb{P}_Y^r$. In this case, locally over $Y$, and possibly replacing $\mathcal{O}_Y(1)$ by a multiple again, the sheaf $R$ is a quotient of $\mathcal{O}_U[y_1, y_2, \ldots, y_r]$, where $y_i = X_i/X_0$ and $X_i$ are homogeneous coordinates on $\mathbb{P}_Y^r$. In particular $R$ is finitely generated.
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