Let $S$ be a graded ring.
$f$ : homogeneous element of degree $> 0$.
$S_f$= localized ring $T^{-1}S$, where $T$={ $f^k|k \ge 0$}.
$S_{(f)}$ = {elements of degree zero in $S_f$}.
Then there is 1-to-1 corresponence between homogeneous prime ideals of $S$ which does not contain $f$ and pime ideals of $S_{(f)}$.
In my opinion, homogeneous prime ideal $P$ in $S$ may relate to pime ideal $(P S_f)\cap S_{(f)}$of $S_{(f)}$.
But I can't show that any prime ideal of $S_{(f)}$ is of the form $(P S_f)\cap S_{(f)}$.