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Let $A=k[x,y,z]$ and $I=(x^2,xy,xz,yz)$. My previous question was how to calculate a primary decomposition of $I$. However there was a part b) added to this exercise, namely to calculate a disassembly for the module $M=A/I$.

If I understand it right, I am supposed to find a chain of submodules $$0=M_0 \subset M_1 \subset\cdots\subset M_n=M$$ such that $M_i/M_{i-1} \cong A/P_i$ where $P_i \in \text{Spec}(A)$.

I'm aware of that $\operatorname{Ass}(A/I)=\{(x,y),(x,z),(x,y,z)\} \subset \{P_1,...,P_n \}$ and I have a feeling this can help me, but I don't know how. Any hints?

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Hint. $0\subset(x,yz)/I\subset(x,y)/I\subset(x,y,z)/I\subset A/I$.

user26857
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  • The first two steps I have already done, that is, $(x,y)/I \subset (x,y,z)/I$, the problem is I couldnt find a prime ideal $P$ such that $((x,y,z)/I)/(x,y)/I \cong A/P$. the same problems occurs as I want to continue, how do you know that $(x,yz)/I$ is the next step? How should one think here? I think I am missing the point, namely the strategy. –  Jan 13 '15 at 15:30