For \(f \in S_d\) homogeneous of positive degree, let \(D_+(f) \da \ts{\mfp \in \Proj S \st f \notin \mfp}\). Then \[ (D_+(f), \ro{\OO_{\Proj S}}{D_+(f)}) \cong \Spec S_{(f)} , \] where \(S_{(f)}\) is the degree-zero part of \(S[\inverseof{f}]\). The \(D_+(f)\) for \(f \in S_+\) cover \(\Proj S\), so \(\Proj S\) is a scheme.
For \(S = \kxnz\) with the usual grading, \(D_+(x_i) \cong \Spec k[x_0/x_i, \dots, x_n/x_i] \cong \AA^n\slice k\), recovering the standard charts of \(\PP^n\slice k\).
The scheme structure is exhibited by the affine charts \(D_+(f)\cong\Spec S_{(f)}\): inverting a homogeneous element and passing to degree zero produces ordinary affine functions. Discarding \(V(S_+)\) is what makes the \(D_+(f)\) a cover, since a prime containing all of \(S_+\) lies in no chart.
Over a base, define \(\PP^n\slice \ZZ \da \Proj \ZZ[x_0, \dots, x_n]\) and then \[ \PP^n\slice S \da \fiberprod{\PP^n\slice \ZZ}{\Spec \ZZ}{S} . \] The same pattern gives \(\AA^n\slice S\) from \(\Spec \ZZ[x_1, \dots, x_n]\). Thus projective space over a base scheme is obtained by base change from the universal projective space over \(\Spec\ZZ\), rather than by choosing new gluing data for each base.
Dependencies
D-SCHPROJ$\Proj$ of a graded ringuses