Proj construction

Definition

Let $S = \bigoplus_{d \ge 0} S_d$ be a commutative graded ring, and let $S_+ = \bigoplus_{d > 0} S_d$. As a set, $\operatorname{Proj} S$ is the set of homogeneous prime ideals $\mathfrak{p}$ of $S$ with $S_+ \not\subseteq \mathfrak{p}$. Its closed sets are the sets $V_+(\mathfrak{a}) = \set{\mathfrak{p} \in \operatorname{Proj} S : \mathfrak{a} \subseteq \mathfrak{p}}$ for homogeneous ideals $\mathfrak{a}$. For a homogeneous $f \in S_+$, let $D_+(f) = \set{\mathfrak{p} \in \operatorname{Proj} S : f \notin \mathfrak{p}}$. Let $S_{(f)}$ be the ring of elements of degree $0$ in the localization $S_f$, that is, the fractions $a / f^n$ with $a$ homogeneous of degree $n \deg f$.

The structure sheaf $\mathcal{O}$ on $\operatorname{Proj} S$ is the unique sheaf of rings such that, for every homogeneous $f \in S_+$:

1. $\mathcal{O}(D_+(f)) = S_{(f)}$;
2. $(D_+(f), \mathcal{O}|_{D_+(f)})$ is isomorphic to the affine scheme $\operatorname{Spec} S_{(f)}$, by $\mathfrak{p} \mapsto \mathfrak{p} S_f \cap S_{(f)}$.

The sets $D_+(f)$ cover $\operatorname{Proj} S$, so $(\operatorname{Proj} S, \mathcal{O})$ is a scheme.