Let $A$ be the power series ring $\mathbb{C}[[x,y]]$.


Assume we are given two ideals $I,J$ of finite length in $A$ such that:


* $xJ\subseteq I\subseteq J$


Is it possible to find ideals of finite length $B,C$ in $A$ such that:


* $xC\subseteq B\subseteq C$
* $B\subset I$ and $I/B\cong R/J$
* $C\subset J$ and $J/C\cong R/I$


Here are my thoughts so far: we see that $x$ annihilates $J/I$ so that this is a module of finite length over $\mathbb{C}[[y]]$, hence also some power of $y$ annihilates $J/I$. I thought maybe we should write $I=<f\_1,\ldots,f\_n>$ and $J=<h\_1,\ldots,h\_m>$ and see if we can use these generators to construct/find the wanted ideals, but this seems quite hard.


Is it even possible to find such ideals $B$ and $C$? Any ideas how to approach this problem are welcome. One could also ask more genrally:


Assume we are given $n$ ideals $I\_r$ of finite length in $A$ such that:


* $xI\_r\subseteq I\_1\subseteq I\_r$ for all $r\geq 2$
* for all $r\geq 2$: $xI\_r\subseteq I\_s\subseteq I\_r$ for all $s\geq r+1$


Is it possible to find ideals of finite length $J\_r\subset I\_r$ in $A$ for $r=1\ldots n$ which satisfy:


* $xJ\_r\subseteq J\_1\subseteq J\_r$ for all $r\geq 2$
* for all $r\geq 2$: $xJ\_r\subseteq J\_s\subseteq J\_r$ for all $s\geq r+1$
* $I\_r/J\_r\cong R/I\_{r-1}$ ($r\geq 2$) and $I\_1/J\_1\cong R/I\_n$



How to find ideals of finite length in a power series ring with special properties?
As I commented, in general, the answer is no. For an easy example, take $I=J=(x,y)^2$. Then you require a surjection $f:J\to R/I$. Letting $f(x^2)=a, f(xy)=b, f(y^2)=c$, since $y.x^2=x.xy$, we get $ya=xb$ and similarly, $yb=xc$, using $y.xy=x.y^2$. Since $ya, xb\in (x,y)/I$, which is a $k$ vector space with basis $x,y$, we see that $a,b\in (x,y)$. Similarly, we get $b,c\in (x,y)$. Thus $a,b,c\in (x,y)/I$ and then the map $f$ can not be onto.


