Algebraic Geometry Question - on ideals of algebraic sets

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SUMMARY

The discussion centers on the relationship between the ideals of algebraic sets in algebraic geometry, specifically addressing the claim that I(X ∩ Y) is not always equal to I(X) + I(Y). The user explores this concept using examples from the polynomial ring \(\mathbb{C}[X]\) and seeks clarification on the conditions under which this equality holds. The suggestion to examine sets that meet tangentially in \(\mathbb{C}[X,Y]\) is provided as a potential avenue for further exploration.

PREREQUISITES
  • Understanding of algebraic sets and their representation as V(J) in algebraic geometry.
  • Familiarity with ideals in the context of polynomial rings, specifically \(\mathbb{C}[X]\) and \(\mathbb{C}[X,Y]\).
  • Knowledge of the intersection and sum of ideals in algebraic geometry.
  • Basic concepts of tangential intersections of algebraic sets.
NEXT STEPS
  • Investigate the properties of ideals in \(\mathbb{C}[X,Y]\) and their implications for algebraic sets.
  • Explore examples of tangential intersections of algebraic sets to understand the conditions affecting ideal equality.
  • Study the concept of primary decomposition in algebraic geometry to gain insights into ideal relationships.
  • Learn about the Nullstellensatz theorem and its applications in relating ideals and algebraic sets.
USEFUL FOR

Mathematicians, algebraic geometers, and students studying algebraic geometry who are interested in the properties of ideals and their relationships with algebraic sets.

slevvio
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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 \in X | f(x) = 0 for all x \in X}. I am trying to show that I(X \cap Y) is not always equal to I(X) + I(Y), so I have tried many examples of ideals of \mathbb{C}[X] but I am not getting anywhere.

Any help would be appreciated!

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

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