Editing Hilbert's Nullstellensatz (section)

== Analytic Nullstellensatz (Rückert’s Nullstellensatz)==
The Nullstellensatz also holds for the germs of holomorphic functions at a point of complex ''n''-space <math>\Complex^n.</math> Precisely, for each open subset <math>U \subseteq \Complex^n,</math> let <math>\mathcal{O}_{\Complex^n}(U)</math> denote the ring of holomorphic functions on ''U''; then <math>\mathcal{O}_{\Complex^n}</math> is a ''[[sheaf (mathematics)|sheaf]]'' on <math>\Complex^n.</math> The stalk <math>\mathcal{O}_{\Complex^n, 0}</math> at, say, the origin can be shown to be a [[Noetherian ring|Noetherian]] [[local ring]] that is a [[unique factorization domain]].

If <math>f \in \mathcal{O}_{\Complex ^n, 0}</math> is a germ represented by a holomorphic function <math>\widetilde{f}: U \to \Complex </math>, then let <math>V_0(f)</math> be the equivalence class of the set 

:<math>\left \{ z \in U \mid \widetilde{f}(z) = 0  \right\},</math>

where two subsets <math>X, Y \subseteq \Complex^n</math> are considered equivalent if <math>X \cap U = Y \cap U</math> for some neighborhood ''U'' of 0. Note <math>V_0(f)</math> is independent of a choice of the representative <math>\widetilde{f}.</math> For each ideal <math>I \subseteq \mathcal{O}_{\Complex^n,0},</math> let <math>V_0(I)</math> denote <math>V_0(f_1) \cap \dots \cap V_0(f_r)</math> for some generators <math>f_1, \ldots, f_r</math> of ''I''. It is well-defined; i.e., is independent of a choice of the generators.

For each subset <math>X \subseteq \Complex ^n</math>, let 

:<math>I_0(X) = \left \{ f \in \mathcal{O}_{\Complex^n,0} \mid V_0(f) \supset X \right \}.</math>

It is easy to see that <math>I_0(X)</math> is an ideal of <math>\mathcal{O}_{\Complex ^n, 0}</math> and that <math>I_0(X) = I_0(Y)</math> if <math>X \sim Y</math> in the sense discussed above.

The '''analytic Nullstellensatz''' then states:<ref>{{harvnb|Huybrechts|loc=Proposition 1.1.29.}}</ref> for each ideal <math>I \subseteq \mathcal{O}_{\Complex ^n, 0}</math>,

:<math>\sqrt{I} = I_0(V_0(I))</math>

where the left-hand side is the [[radical of an ideal|radical]] of ''I''.<!-- TODO: need to mention an analytic subset, which apparently is yet to be created. -->