Projective space and the Proj construction

Write S as a direct sum of nonnegative graded pieces and let S_+ be the ideal of positive-degree elements. Proj S consists of homogeneous prime ideals that do not contain S_+. Its closed sets come from homogeneous ideals. A standard open D_+(f), where f is homogeneous of positive degree, consists of primes avoiding f. These opens cover Proj and intersect by multiplying their defining homogeneous elements. The excluded ideal reflects the fact that all homogeneous coordinates cannot vanish simultaneously.

For a polynomial ring A[X₀,…,Xₙ] with each variable in degree one, these standard charts build projective n-space over A. Over a field its rational points have the familiar nonzero-coordinate description modulo common scalar multiplication. That description does not exhaust its scheme points or describe all families over arbitrary test rings. In particular, a collection of homogeneous coordinates over a ring requires appropriate local generation conditions; a field-based quotient picture is not a replacement for the scheme construction.

If f has degree d, the degree-zero fractions are represented by g/f^r with g homogeneous of degree rd. The numerator and denominator have equal total degree, so the fraction is unchanged by homogeneous scaling. In ordinary projective space the chart where X_i is nonzero has coordinates X_j/X_i, giving a polynomial ring in n variables over the base. Overlaps are ordinary localizations of these chart rings, which is what allows the structure sheaves to glue.

If S is generated by finitely many degree-one elements over S₀, a homogeneous presentation realizes Proj S as a closed subscheme of a finite-dimensional projective space over S₀. In this setting the sheaf O(1) is invertible and is the hyperplane line bundle. Arbitrary positive gradings still allow Proj, but these conclusions cannot simply be transferred unchanged: degree-one generators may be absent, and a chosen twist need not be an invertible sheaf. The grading hypotheses must travel with every statement about charts, twists, or embeddings.

A projective closed subscheme defined by a homogeneous ideal is computed chart by chart by dividing its equations by matching powers of the chosen nonvanishing coordinate. This dehomogenization retains the quotient-ring structure, including possible nilpotents. Passing to another chart requires checking the transition maps, rather than merely joining drawings of the affine zero sets. A point omitted by one chart may be a perfectly ordinary point of the projective scheme, often interpreted as lying at infinity for that affine description.

The smooth conic provides a useful example in which quadratic homogeneous coordinates describe a projective line. Its local affine coordinate is a ratio, and another ratio supplies the reciprocal chart. This does not make every plane curve a projective line: the equation and transition maps are doing substantive work. More broadly, different graded presentations can lead to isomorphic underlying schemes while specifying different polarizations. Later Hilbert polynomials will depend on the chosen ample line bundle, not only on the abstract space.