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Let $k$ be a field and $S=k[x,y,z]$. Let $m=(x,y,z)$ and $J,K\subseteq m$ be proper homogeneous ideals in $S$. Is this true that we always have:

$$[JK:(x)][JK:(y,z)]\subseteq JK \ ?$$

Some background: the case $K=m$ was asked at Is $[Im:(x)][Im:(y,z)]\subseteq Im$ in $k[x,y,z]$?. It was not so hard. One can prove that the result holds if $J,K$ are monomial or if $K\subset m^2$ is integrally closed. The statement is related to the question of whether the product of two ideals is Golod. See the preprint Dao and De Stefani - On monomial Golod ideals for some partial results and why a modest monetary reward is offered for this frustrating problem (Question 4.6). My personal guess is that this is probably true in char. $0$.

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  • $\begingroup$ Your equation only contains the product $JK$; the ideals $J$ and $K$ never appear separately. Is this intentional? $\endgroup$
    – David E Speyer
    Commented Feb 5, 2019 at 16:48
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    $\begingroup$ @DavidESpeyer: I am not sure what you meant. One can wrote, let $I=JK$, but being product is crucial for this property to have a hope of being true. $\endgroup$
    – Hailong Dao
    Commented Feb 5, 2019 at 17:13
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    $\begingroup$ Thanks. I've looked up the paper now, and I see that you did mean what you wrote. $\endgroup$
    – David E Speyer
    Commented Feb 5, 2019 at 17:28

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