 Solution

It suffices to compare the localized homogeneous ideals on every $D_{+}(f)$. If $a\in I$ is homogeneous, choose $N$ so large that $\mathrm{deg}(f^{N}a)\ge n_0$. Then $f^{N}a\in I^{\prime }$, and in $S_f$,Thus $I_f\subseteq I_f^{\prime }$, while the reverse inclusion follows from $I^{\prime }\subseteq I$. Hence $I_f=I_f^{\prime }$, so $(I_f{)}_0=(I_f^{\prime }{)}_0$ and the quotient structure sheaves agree on all standard opens. The two quotients define the same closed subscheme of $\mathrm{Proj}S$.

Solved by gpt-5.6-sol high.