Though we cannot evaluate polynomials \(F\) at points of \(\mathbb{P}^n\), it makes sense to talk about the zeroes of \(F\). Observe that for \(|k| = \infty \), if \(F = \sum F_i \) is a decomposition of \(F\) into homogeneous polynomials of degree \(i\) then for all \( \underline{\alpha} \in \mathbb{P}^n\) we have
$$F(\underline{\alpha}) = 0 \iff F_i (\underline{\alpha}) = 0 \forall i.$$
Proof. We must have \(F(\lambda \underline{\alpha}) = 0 \forall \lambda \in k\). Now then RHS, viewed as a polynomial in \(\lambda\) has infinitely many roots.
This shows in particular, that the following two definitions of a homogeneous ideal \(I\) of \(k[x_1, \ldots, x_n ]\) are equivalent:
- \(I\) is generated by homogeneous polynomials
- if \(F \in I\) then so are all of its homogeneous components \(F_i\).
As in the affine case, we still have a correspondence between nonempty projective algebraic sets in \(\mathbb{P}^n\) with homogeneous, radical, relevant ideals of \(k[x_1, \ldots, x_n ]\). Under this correspondence, irreducible projective subsets still correspond to prime ideals; however, points no longer correspond to max ideals.
In the case of a projective \(X \subset \mathbb{P}^n\), we again have a correspondence between its projective algebraic subsets and the ideals of its homogeneous coordinate rings. However, unlike the affine case, this homogeneous coordinate ring is dependent on the specific embedding \( X \hookrightarrow \mathbb{P}^n\) and even for the same embedding, there might be several graded rings structure associated to it. Furthermore, again, unlike the affine case, its elements aren't functions on \(X\).
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