Affine Varieties and the Vanishing Ideal

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In summary, the conversation discusses the concepts of affine varieties and vanishing ideals in the realm of polynomial rings. It is explained that the vanishing ideal is a subset of the affine variety and that there is a one-to-one correspondence between affine varieties and prime ideals in the polynomial ring. It is also mentioned that ideals can be used to compute with affine varieties.
  • #1
Szichedelic
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Can someone please explain to me these two concepts and how they differ from each other? I'm taking a class entitled Math & Computers which emphasizes algebraic geometry in a symbolic-computational setting. That being said, the class is not very oriented towards explaining and understanding these ideas as it is assumed that I've already had experience in dealing with them.

Most specifically, I'm talking about affine varieties and vanishing ideals in the realm of polynomial rings. Is the vanishing ideal a subset of the affine variety?
 
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  • #2
If you have a subset Z of your affine space (for example Rn), the corresponding vanishing ideal, is the set of polyomials in n variables being zero on every point of Z. If you have a set T of polynomials, you can talk about the common zeros of all polynomials in T, which will be a subset of Rn, and call this an affine variety, usually with some irreducibility-condition, depending on your text-book.

So loosely speaking, your objects, varieties and ideals, live in two different worlds, Rn and the polynomial ring, respectively. But a variety gives an ideal, and an ideal gives a variety. To be more precise, you can prove a theorem showing a one-to-one correspondence between the (irreducible) affine varieties in Rn, and prime ideals in the polynomial ring, and by using this theorem you can translate statements about ideals to statements about varieties, and vice versa.
 
  • #3
I had a moment of insight last night around 1am. I understand now how ideals and affine varieties are linked. The vanishing ideal is the ideal of a variety. Ideals also give us a way to compute with affine varieties, correct?
 
  • #4
varieties consist of points where certain polynomials, the ones in the vanishing ideal, vanish.

"vanish" means equal zero.
 
  • #5


Affine varieties and vanishing ideals are fundamental concepts in algebraic geometry, a branch of mathematics that studies solutions to polynomial equations. In simple terms, an affine variety is a set of points in n-dimensional space that satisfy a system of polynomial equations, while a vanishing ideal is a set of polynomials that vanish (or evaluate to zero) on a given affine variety.

To better understand the relationship between these two concepts, it is helpful to think of affine varieties as the "solution sets" and vanishing ideals as the "equations" that define them. For example, consider the affine variety defined by the equation x^2 + y^2 = 1. This is the familiar unit circle in the xy-plane. The vanishing ideal for this variety would be the set of all polynomials that vanish on the unit circle, such as x^2 + y^2 - 1.

One key difference between affine varieties and vanishing ideals is that while an affine variety is a geometric object, a vanishing ideal is an algebraic object. This means that affine varieties can be visualized and studied geometrically, while vanishing ideals are manipulated symbolically using algebraic techniques.

In terms of their relationship, the vanishing ideal of an affine variety is a subset of the polynomial ring defining that variety. In the previous example, the vanishing ideal is a subset of the polynomial ring in two variables, x and y. This is because the vanishing ideal is composed of polynomials in x and y that vanish on the points of the unit circle.

I hope this explanation helps clarify the concepts of affine varieties and vanishing ideals for you. As you continue to study algebraic geometry, you will encounter these concepts in various contexts and gain a deeper understanding of their connections and applications.
 

1. What is an affine variety?

An affine variety is a subset of n-dimensional space that can be defined as the common solutions to a set of polynomial equations in n variables. It can also be described as the zero locus of a set of polynomials.

2. What is the vanishing ideal of an affine variety?

The vanishing ideal of an affine variety is the set of polynomials that vanish at every point of the variety. In other words, it is the ideal generated by the polynomials that define the variety.

3. What is the relationship between affine varieties and algebraic sets?

An affine variety is a type of algebraic set, which is a subset of n-dimensional space defined by polynomial equations. However, not all algebraic sets are affine varieties, as some may have additional structure or constraints.

4. How are affine varieties used in algebraic geometry?

Affine varieties are a fundamental object of study in algebraic geometry. They provide geometric interpretations of algebraic concepts and allow for the study of algebraic structures and equations in a geometric framework.

5. How are affine varieties classified?

Affine varieties can be classified by their dimension, which is the number of variables in the defining polynomial equations. They can also be classified by their degree, which is the maximum degree of the polynomials in the defining equations.

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