algebraic sets and polynomial ideals (original) (raw)

Suppose k is a field. Let 𝔸kn denote affine n-space over k.
For S⊆k⁢[x1,…,xn], define V⁢(S), the zero set of S, by

V⁢(S)={(a1,…,an)∈kn∣f⁢(a1,…,an)=0⁢ for all ⁢f∈S}

We say that Y⊆𝔸kn is an (affine) algebraic set if there exists T⊆k⁢[x1,…,xn] such that Y=V⁢(T). Taking these subsets of 𝔸kn as a definition of the closed sets of a topology induces the Zariski topologyMathworldPlanetmath over 𝔸kn.
For Y⊆𝔸kn, define the deal of Y in k⁢[x1,…,xn] by

I⁢(Y)={f∈k⁢[x1,…,xn]∣f⁢(P)=0⁢ for all ⁢P∈Y}.

It is easily shown that I⁢(Y) is an ideal of k⁢[x1,…,xn].
Thus we have defined a function V mapping from subsets of k⁢[x1,…,xn] to algebraic sets in 𝔸kn, and a function I mapping from subsets of 𝔸n to ideals of k⁢[x1,…,xn].
We remark that the theory of algebraic sets presented herein is most cleanly stated over an algebraically closed field. For example, over such a field, the above have the following properties:

    1. S1⊆S2⊆k⁢[x1,…,xn] impliesV⁢(S1)⊇V⁢(S2).
    1. Y1⊆Y2⊆𝔸kn impliesI⁢(Y1)⊇I⁢(Y2).
    1. For any ideal 𝔞⊂k⁢[x1,…,xn],I⁢(V⁢(𝔞))=Rad⁡(𝔞).
    1. For any Y⊂𝔸kn, V⁢(I⁢(Y))=Y¯, the closureMathworldPlanetmathof Y in the Zariski topology.