Is there any example of a homogeneous (not monomial) $(x,y)$-primary ideal $I$ in $K[x,y]$ such that $I$ is complete and and there exists a minimal reduction $J$ of $I$ such that $J\overline{I^n}=\overline{I^{n+1}}$ for all $n\geq 1.$
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$\begingroup$ Can you define these terms? What is a complete ideal? $\endgroup$– Will SawinCommented Jul 14, 2015 at 14:45
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$\begingroup$ An ideal is complete means it is integrally closed. $\endgroup$– CuspCommented Jul 14, 2015 at 15:53
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$\begingroup$ en.wikipedia.org/wiki/Integral_closure_of_an_ideal $\endgroup$– CuspCommented Jul 14, 2015 at 15:53
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As long as $K$ is a field, you can refer to the following result of Zariski:
(Zariski) Let $R$ be a $2$ dimensional regular local ring. Then the product of integrally closed ideals is integrally closed.
Therefore, choose say $I = (x+y, y^2)$. This is integrally closed: The length of the module $R/I$ is two and $I \subseteq (x,y)R$. If it were not integrally closed, then the maximal ideal $m = (x,y)R$ is its integral closure. Then $Im^q = m^{q+1}$ for $q \gg 0$. However, the LHS does not contain $x^{q+1}$.
Since $I$ is a complete intersection, it is its own minimal reduction. Therefore, $$ \overline{I^{n+1}} = I^{n+1} = I I^n = I \overline{I^n}, $$ where the first and last equalities follow from the result of Zariski.
