Hilbert's Nullstellensatz
Projective Nullstellensatz

We can formulate a certain correspondence between homogeneous ideals of polynomials and algebraic subsets of a projective space, called the projective Nullstellensatz, that is analogous to the affine one. To do that, we introduce some notations. Let R = k. {\displaystyle R=k.} The homogeneous ideal,

is called the maximal homogeneous ideal (see also irrelevant ideal). As in the affine case, we let: for a subset S ⊆ P n {\displaystyle S\subseteq \mathbb {P} ^{n}} and a homogeneous ideal I of R,

By f = 0 on S {\displaystyle f=0{\text{ on }}S} we mean: for every homogeneous coordinates (a 0: ⋯: a n) {\displaystyle (a_{0}:\cdots:a_{n})} of a point of S we have f (a 0, …, a n) = 0 {\displaystyle f(a_{0},\ldots,a_{n})=0}. This implies that the homogeneous components of f are also zero on S and thus that I P n (S) {\displaystyle \operatorname {I} _{\mathbb {P} ^{n}}(S)} is a homogeneous ideal. Equivalently, I P n (S) {\displaystyle \operatorname {I} _{\mathbb {P} ^{n}}(S)} is the homogeneous ideal generated by homogeneous polynomials f that vanish on S. Now, for any homogeneous ideal I ⊆ R + {\displaystyle I\subseteq R_{+}}, by the usual Nullstellensatz, we have:

and so, like in the affine case, we have:
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