Editing Complex projective space (section)

===Structure as a scheme===
Another construction of '''CP'''<sup>''n''</sup> (and its Zariski topology) is possible.  Let ''S''<sub>+</sub>&nbsp;⊂&nbsp;''S'' be the [[ideal (ring theory)|ideal]] spanned by the homogeneous polynomials of positive degree:
:<math>\bigoplus_{n>0}S_n.</math>
Define [[Proj|Proj ''S'']] to be the set of all [[homogeneous ideal|homogeneous]] [[prime ideal]]s in ''S'' that do not contain ''S''<sub>+</sub>.  Call a subset of Proj ''S'' closed if it has the form
:<math>V(I) = \{ p\in \operatorname{Proj} S\mid p\supseteq I\}</math>
for some ideal ''I'' in ''S''.  The complements of these closed sets define a topology on Proj ''S''.  The ring ''S'', by [[localization of a ring|localization at a prime ideal]], determines a [[sheaf (mathematics)|sheaf]] of [[local ring]]s on Proj ''S''.  The space Proj ''S'', together with its topology and sheaf of local rings, is a [[scheme (mathematics)|scheme]].  The subset of closed points of Proj ''S'' is homeomorphic to '''CP'''<sup>''n''</sup> with its Zariski topology.  Local sections of the sheaf are identified with the [[rational function]]s of total degree zero on '''CP'''<sup>''n''</sup>.