Proj of a graded ring

Let $S=\bigoplus_{d\ge 0}S_d$ be a commutative graded ring, and let

S_+:=\bigoplus_{d>0}S_d

be its irrelevant ideal. The space $\operatorname{Proj}S$ consists of the homogeneous prime ideals of $S$ that do not contain $S_+$ .

For a homogeneous ideal $I\subseteq S$ , the sets

V_+(I):=\{\mathfrak p\in\operatorname{Proj}S:I\subseteq\mathfrak p\}

are the closed sets of its topology. If $f\in S$ is homogeneous of positive degree, the standard open subset

D_+(f):=\{\mathfrak p:f\notin\mathfrak p\}

D_+(f)\cong\operatorname{Spec}(S_f)_0,

where $(S_f)_0$ is the degree-zero part of the localization $S_f$ . These affine pieces determine the structure sheaf on $\operatorname{Proj}S$ .

For the polynomial ring with its standard grading,

\operatorname{Proj}k[x_0,\ldots,x_n]=\mathbb P_k^n,