Dtyjlui

Ravi Vakil 9.2.B is "Suppose that $S rightarrow R$ is a surjection of graded rings. Show that the induced morphism $text{Proj }R rightarrow text{Proj }S$ is a closed embedding."

I don't even see how to prove that the morphism is affine. The only ways I can think of to do this are to either classify the affine subspaces of Proj S, or to prove that when closed morphisms are glued, one gets a closed morphism.

Are either of those possible, and how can this problem be done?

I think a good strategy could be to verify the statement locally, and then verify that the glueing is successful, as you said. Let us call $phi:Sto R$ your surjective graded morphism, and $phi^ast:textrm{Proj},,Rto textrm{Proj},,S$ the corresponding morphism. Note that $$textrm{Proj},,R=bigcup_{tin S_1}D_+(phi(t))$$
because $S_+$ (the irrelevant ideal of $S$) is generated by $S_1$ (as an ideal), so $phi(S_+)R$ is generated by $phi(S_1)$. For any $tin S_1$ you have a surjective morphism
$S_{(t)}to R_{phi(t)}$ (sending $x/t^nmapsto phi(x)/phi(t)^n$, for any $xin S$), which corresponds to the canonical closed immersion of affine schemes $phi^ast_t:D_+(phi(t))hookrightarrow D_+(t)$. It remains to glue the $phi^ast_t$'s.

Almost 7 years late! Here is my try. Hallo Thorsten!

I call our maps $f colon operatorname{Proj} B to operatorname{Proj} A$ and $varphi colon A to B$. Surjectivity implies that we actually have a well-defined map $operatorname{Proj} B to operatorname{Proj} A$ and a morphism of schemes in this way.

Being a closed immersion is affine-local on the target. Therefore we can consider some cover of open affines $bigcup_{j in J} V_j = operatorname{Proj} A$ and then check that for each $j in J$ we have a closed immersion $f mid_{f^{-1}(V_j)} colon f^{-1}(V_j) hookrightarrow V_j$. This is described in Vakil's notes as an exercise.

We have that the collection over all homogeneous $g in A$ of $D(g) = {,p in operatorname{Proj} A mid g notin p ,}$ cover $operatorname{Proj} A$. As $varphi$ is surjective, we have $f^{-1} (D(g)) = D(varphi(g))$.
We now have
begin{align*}
f mid_{D(varphi(g))} colon D(varphi(g)) & hookrightarrow D(g) \
p & mapsto varphi^{-1} (p) , .
end{align*}

These sets are all open affines! For any graded ring $R$, we have for any homogeneous $h in R$ the identification $D(h) = operatorname{Spec}(R_h)_0 = operatorname{Spec}{, frac{x}{h^n} mid n in mathbb N, , deg x = deg h cdot n ,}$. (Sometimes, $(R_h)_0$ is confusingly written as $R_{(h)}$.) Our map can then be seen as
begin{align*}
f mid_{operatorname{Spec} (B_{varphi(g)})_0} colon operatorname{Spec} (B_{varphi(g)})_0 & hookrightarrow operatorname{Spec} (A_g)_0 \
p & mapsto varphi^{-1} (p) ; ,
end{align*}
which corresponds to the surjective ring homomorphism
begin{align*}
varphi (D(g)) colon (A_g)_0 & to (B_{varphi(g)})_0 \
frac{x}{g^n} & mapsto frac{varphi(x)}{varphi(g)^n} ; ,
end{align*}
which means that $f mid_{f^{-1}(D(g))} colon f^{-1}(D(g)) hookrightarrow D(g)$ is a closed immersion, concluding that $f$ is a closed immersion.