# The structure sheaf on $\operatorname{Proj}S$

Suppose $$S$$ is a $$\mathbb{Z}_{\geq 0}$$ graded ring that that $$f\in S_+$$ where $$S_+$$ is the irrelevant ideal. I know that we may associate $$D_+(f)$$ with the set of prime ideal of the zero degree component of S. That is, $$D(f)=\operatorname{Spec}(S_f)_0$$. Further, we define the set $$\operatorname{Proj}S$$ to be the set of homogeneous prime ideals that do not contain the irrelevant ideal $$S_+$$.

I am wondering why the following exercise (4.5.k) in Vakil's algebraic geometry notes implies what the structure of $$\operatorname{Proj}S$$ should be: "If $$f,g\in S_+$$ are homogeneous and nonzero, describe an isomorphism between $$\operatorname{Spec}(S_{fg})_0$$ and the distinguished open subset $$D(g^{\deg f}/f^{\deg g})$$ of $$\operatorname{Spec}(S_f)_0$$. Similarly, $$\operatorname{Spec}(S_{fg})_0$$ is identified with a distinguished open subset of $$\operatorname{Spec}(S_g)_0$$. We then glue the various $$\operatorname{Spec}(S_f)_0$$ (as $$f$$ varies) altogether, using these pairwise gluings.

The author then suggests that this exercise implies what the structure sheaf on $$\operatorname{Proj}S$$ should look like. If my intuition is correct, this is because using the identifications above, and the fact that the $$D_+(f)$$, where $$f\in S_+$$, form a base for the topology on $$\operatorname{Proj}S$$, we can use the identification above to glue $$D(f)$$ and $$D(g)$$ along $$D(fg)$$, and then use the structure sheaf for the gluing of affine scheme. Is this intuition correct?

Yes, that's correct. What's happening here is that we have a cover of $$\operatorname{Proj} S$$ by sets of the form $$\operatorname{Spec}(S_f)_0$$, each of which comes equipped with the usual affine structure sheaf.
To glue these into a sheaf on all of $$\operatorname{Proj} S$$, we need to describe sheaf isomorphisms $$\phi_{fg}$$ defined on the pairwise intersections. But we also need a compatibility condition on the triple intersections, something like $$\phi_{gh} \circ \phi_{fg} = \phi_{fh}$$.
• A small formatting tip: using \operatorname instead of \rm produces better formatting for things like $\operatorname{Spec}$. I've made this upgrade in your post. Sep 15, 2020 at 20:14