In this exercise we have a surjective graded ring homomorphism $\varphi:S\to T$. This induces a morphism $f:$Proj$(T)\to $Proj$(S)$ by contraction of ideals. I'm asked to show that $f$ is a closed immersion.
In orther to show that $f^\sharp:\mathcal{O}_{Proj(S)}\to f_*\mathcal{O}_{Proj(T)}$ is surjective, I restric it to an open subset of the form $D_+(g)=\{p\in$ Proj$(S)\mid g\notin p\}$. Surjectivity in these sets would imply surjectivity on the fibers so I'd be done.
I would like to show that $f_*\mathcal{O}_{Proj(T)}(D_+(g))=\mathcal{O}_{Proj(T)}(D_+(\varphi(g)))$ to apply proposition 2.5 of Hartshorne and get a natural homomorphism $S_{(g)}\to T_{(\varphi(g))}$, which is clearly surjective.
To show this, it is enough to show $f^{-1}(D_+(g))=D_+(\varphi(g))$. Explicitily, the firs set, using that $f^{-1}$ is given by ideal extension and that $\varphi$ is surjective, is $\{\varphi(p)\in$Proj$(T)\mid \varphi(g)\notin\varphi(p)\}$. So what I need to show is that if $g\notin p$, then $\varphi(g)\notin \varphi(p)$. This is obviously false for a general surjective ring homomorphism, but I guess this is true for the graded case.
How can I show this final implication (if it is true)? If it is not true, how can I solve the exercise?
I think it is better to use the adjunction relation from exercise 1.8, which tells us that it is equivalent to prove surjectivity of the homomorphism $\mathcal{O}_{Proj(S),\varphi^{-1}(p)}\to\mathcal{O}_{Proj(T),p}$ (using that $f^{-1}G_x=G_{f(x)}$ for a sheaf $G$). Then it is possible to apply proposition 2.5a), which gives us the homomorphism $S_{(\varphi^{-1}(p))}\to T_{(p)}$.