Nonlinear Equations - UFRJ

26 [CH. 2: THE NULLSTELLENSATZ
To each ideal J of polynomials, we associated its zero set
Z(J) = {x ∈ k n : ∀f ∈ J, f(x) = 0}.
Those two operators are inclusion reversing:
If X ⊆ Y then I(Y ) ⊆ I(X).
If J ⊆ K then Z(K) ⊆ Z(J).
Hence, compositions Z ◦ I and I ◦ Z are inclusion preserving:
If X ⊆ Y then (Z ◦ I)(X) ⊆ (Z ◦ I)(Y ).
If J ⊆ K then (I ◦ Z)(J) ⊆ (I ◦ Z)(K).
By construction, compositions are nondecreasing:
X ⊆ (Z ◦ I)(X) and J ⊆ (I ◦ Z)(J).
The operation Z ◦ I is called Zariski closure. It has the following
property. Suppose that X is Zariski closed, that is X = Z(J) for
some J. Then
(Z ◦ I)(X) = X.
Indeed, assume that x ∈ (Z ◦ I)(X). Then for all f ∈ I(X),
f(x) = 0. In particular, this holds for f ∈ J. Thus x ∈ X.
The opposite is also true. Suppose that J = I(X). We claim that
I(Z(J)) = J.
Indeed, let f ∈ I(Z(J)). This means that f vanishes in all of Z(J).
In particular it vanishes in X ⊆ Z(J). So f ∈ J = I(X).
The operation I ◦Z is akin to the closure of a set, but more subtle.
Example 2.25. Let n = 1 and a ∈ k. Let J = ((x − a) 3 ) be the
ideal of polynomials vanishing at a with multiplicity ≥ 3. Then,
Z(J) = {a} and I(Z(J)) = ((x − a)) the polynomials vanishing at a
(no multiplicity assumed).