algebraic sets and polynomial ideals

Suppose k is a field. Let 𝔸kn denote affine n-space over k.
For SβŠ†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 YβŠ†π”Έkn is an (affine) algebraic set if there exists TβŠ†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 YβŠ†π”Έkn, 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. 1.

    S1βŠ†S2βŠ†k⁒[x1,…,xn] implies V⁒(S1)βŠ‡V⁒(S2).

  2. 2.

    Y1βŠ†Y2βŠ†π”Έkn implies I⁒(Y1)βŠ‡I⁒(Y2).

  3. 3.

    For any ideal π”žβŠ‚k⁒[x1,…,xn], I⁒(V⁒(π”ž))=Rad⁑(π”ž).

  4. 4.

    For any YβŠ‚π”Έkn, V⁒(I⁒(Y))=YΒ―, the closureMathworldPlanetmath of Y in the Zariski topology.

From the above, we see that there is a 1-1 correspondence between algebraic sets in 𝔸kn and radical ideals of k⁒[x1,…,xn]. Furthermore, an algebraic set YβŠ†π”Έkn is an affine varietyMathworldPlanetmath if and only if I⁒(Y) is a prime idealMathworldPlanetmathPlanetmathPlanetmath. As an example of how things can go wrong, the radical ideals (1) and (x2+1) in ℝ⁒[x] define the same zero locus (the empty setMathworldPlanetmath) inside of ℝ, but are not the same ideal, and hence there is no such 1-1 correspondence.

Title algebraic sets and polynomial ideals
Canonical name AlgebraicSetsAndPolynomialIdeals
Date of creation 2013-03-22 13:05:40
Last modified on 2013-03-22 13:05:40
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 16
Author mathcam (2727)
Entry type Definition
Classification msc 14A10
Synonym vanishing set
Related topic Ideal
Related topic HilbertsNullstellensatz
Related topic RadicalOfAnIdeal
Defines zero set
Defines algebraic set
Defines ideal of an algebraic set
Defines affine algebraic set