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I am looking for a counterexample in order to answer to the following:

Is the intersection of two closed irreducible sets in $\mathbb A_{\mathbb C}^3$ still irreducible?

The topology on $\mathbb A_{\mathbb C}^3$ is clearly the Zariski one; by irreducible set, I mean a set which cannot be written as a union of two proper closed subsets (equivalently, every open subset is dense).

I think the answer to the question is "No", but I do not manage to find a counterexample. I think I would be happy if I found two prime ideals (in $\mathbb C[x,y,z]$) s.t. their sum is not prime. Am I right? Is there an easier way?

Thanks.

Romeo
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1 Answers1

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Choose any two irreducible plane curves, they will intersect in a finite number of points.

Damien L
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  • Thanks for the answer. I know it can be a silly question, but since I am a beginner in Algebraic Geometry, I have to ask: may I consider for example the curves $x^{2}+y^{3}=1$ and the $x$-axis? I'm working in $\mathbb A^2$ since you suggest of working with plane curves, but I think there will be no problem for $\mathbb A^n$, just embedding the plane (I will identify $\mathbb A^2$ with $z=0$ in $\mathbb A^3$). Am I right? Thanks again. – Romeo Mar 17 '13 at 18:11
  • Your example is right. – Damien L Mar 17 '13 at 18:13
  • Nitpick: Not any two irreducible plane curves, because there is one exception in which you will get an irreducible intersection (and it would arguably be the first example of two irreducible plane curves that one should write down). – Michael Joyce Mar 17 '13 at 18:44