
prime ideal
Let $\displaystyle S$ be a graded ring.
$\displaystyle f$ : homogeneous element of degree $\displaystyle > 0$.
$\displaystyle S_f$= localized ring $\displaystyle T^{1}S$, where $\displaystyle T$={$\displaystyle f^kk \ge 0$}.
$\displaystyle S_{(f)}$ = {elements of degree zero in $\displaystyle S_f$}.
Then there is 1to1 corresponence between homogeneous prime ideals of $\displaystyle S$ which does not contain $\displaystyle f$ and pime ideals of $\displaystyle S_{(f)}$.
In my opinion, homogeneous prime ideal $\displaystyle P$ in $\displaystyle S$ may relate to pime ideal $\displaystyle (P S_f)\cap S_{(f)}$of $\displaystyle S_{(f)}$.
But I can't show that any prime ideal of $\displaystyle S_{(f)}$ is of the form $\displaystyle (P S_f)\cap S_{(f)}$.