(The homogeneous spectrum)
Let \(S = \bigoplus_{d \geq 0} S_d\) be a graded ring and \(S_+ \da \bigoplus_{d \geq 1} S_d\) the irrelevant ideal. Then \(\Proj S\) is the set of homogeneous primes \(\mfp\) with \(\mfp \not\supseteq S_+\), topologised by the closed sets \(V(\mfa) = \ts{\mfp \st \mfp \supseteq \mfa}\) for \(\mfa\) homogeneous. Its structure sheaf sends \(U\) to the functions \(s: U \to \coprod_{\mfp \in U} S_{(\mfp)}\) with \(s(\mfp) \in S_{(\mfp)}\) and locally of the form \(a/f\) with \(a, f\) homogeneous of the same degree and \(f \notin \mfp\), where \(S_{(\mfp)}\) is the degree-zero part of the homogeneous localisation.
Let \(k\) be an algebraically closed field and \(S=k[x_0,\ldots,x_n]\). The closed points of \(\Proj S\) are the homogeneous primes generated by the linear forms vanishing on a line through the origin of \(k^{n+1}\), so they correspond to the points of \(\PP^n(k)\). For homogeneous \(f\) of degree \(d\) and \(v\in k^{n+1}\sm\ts{0}\), \(f(\lambda v)=\lambda^df(v)\), so whether \(f\) vanishes on the line \(kv\) does not depend on \(v\). For \(a,f\) homogeneous of the same degree \(d\) and \(f(v)\ne0\), \(a(\lambda v)/f(\lambda v)=a(v)/f(v)\), so \(a/f\) has a well-defined value at the point \([v]\). The irrelevant ideal \(S_+\) is the ideal of the origin of \(k^{n+1}\) and the only homogeneous prime containing \(S_+\); it contains the ideal of every line through the origin and corresponds to no point of \(\PP^n(k)\).