Algebraic Geometry Question - on ideals of algebraic sets

In summary, the person is seeking advice on a problem involving algebraic sets X and X', with corresponding ideals I(X) and I(Y). They are attempting to prove that the intersection of these sets, I(X ∩ Y), is not always equal to the sum of the ideals, I(X) + I(Y). They have tried various examples in C[X] but have not made any progress. They are asking for assistance and suggestions.
  • #1
slevvio
8
0
Hello everyone, I was wondering if I could get some advice for the following problem:

I have two algebraic sets X, X', i.e. X = V(J), Y = V(J'), and let I(X),I(Y) be the ideals of these sets, i.e. I(X) ={x [itex]\in[/itex] X | f(x) = 0 for all x [itex]\in X[/itex]}. I am trying to show that I(X [itex]\cap[/itex] Y) is not always equal to I(X) + I(Y), so I have tried many examples of ideals of [itex]\mathbb{C}[X][/itex] but I am not getting anywhere.

Any help would be appreciated!

Thanks
 
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  • #2
look in C[X,Y], and try a couple of sets that meet tangentially.
 

1) What is algebraic geometry?

Algebraic geometry is a branch of mathematics that studies the relationship between algebraic equations and geometric shapes. It uses algebraic techniques to study the properties of geometric objects such as curves, surfaces, and higher-dimensional varieties.

2) What is an algebraic set?

An algebraic set is a set of points in n-dimensional space that satisfy a system of polynomial equations. In algebraic geometry, algebraic sets are the fundamental objects of study and can be described using the language of ideals and varieties.

3) What is an ideal in algebraic geometry?

In algebraic geometry, an ideal is a set of polynomials that vanishes at a given point or set of points. Ideals are used to define algebraic sets and their properties, such as dimension and singularities.

4) What is the connection between ideals and algebraic sets?

Ideals and algebraic sets are closely related in algebraic geometry. An ideal defines an algebraic set, and conversely, an algebraic set can be described as the zero locus of an ideal. Ideals also play a crucial role in the study of the geometric properties of algebraic sets.

5) How is algebraic geometry used in other fields of science?

Algebraic geometry has applications in various fields of science, including physics, computer vision, and cryptography. In physics, it is used to study the geometric properties of physical systems, while in computer vision, it is used to develop algorithms for object recognition and image processing. In cryptography, algebraic geometry is used to create secure codes and data encryption methods.

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