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Let $R = k[x, y] $ and $Q = k[x, y]_{(x,y)}$, the localization of $k[x, y]$ at $(0, 0)$. Let $I$ be an ideal of $Q$ generated by a regular sequence of length $2$. Assume additionally $I$ is $(x, y)$-primary, i.e., $(x, y)^n \subset I \subset (x, y)$ for some $n$.

Is it possible to find a regular sequence in $R$, say $a, b$, such that $I = Qa + Qb$, and $I \cap R = Ra + Rb$?

For example, let $I = (x^2 - y^3)Q + (x^3 - y^2)Q$. Easy to check $x^2, y^2\in I$ and $I \cap R = x^2 R + y^2 R$ (but $I \cap R \neq (x^2 - y^3)R + (x^3 - y^2)R$).


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Clearly the number of generators of $I \cap R$ must be greater than 1 by the Principle ideal theorem and $I \cap R$ is $(x, y)$-primary in $R$. Here I think I actually ask if the minimal number of generators of $I \cap R$ is always two. I couldn't find a counterexample. – Alex Miller Jun 24 '12 at 9:08
Maybe I should have said the field k is algebraically closed. – Alex Miller Jun 24 '12 at 20:21

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