On the primary decomposition of an ideal

Question

Let $I$ be the ideal generated by $x^2-y^3$ and $y^2-x^3$ in $\mathbb{C}[x,y]$. I am trying to answer two questions:

1. What is the length of the $\mathbb C[x,y]$-module $\mathbb{C}[x,y]/I$?

2. What is the minimal primary decomposition of $I$?

For question 2, I think one can find the primary decomposition using computer, but I want to learn how to find that decomposition.

Answer 1

2. Let $I=(X^2-Y^3,X^3-Y^2)$.

First one uses the following isomorphism: $\mathbb C[X,Y]/(X^2-Y^3)\simeq\mathbb C[t^2,t^3]$ given by $X\mapsto t^3$ and $Y\mapsto t^2$. The ideal $I/(X^2-Y^3)$ corresponds by this isomorphism to the ideal $(t^9-t^4)$ in $\mathbb C[t^2,t^3]$. But $(t^9-t^4)=(t^4)\cap(t^5-1)$. Then $$I=(Y^2,X^2-Y^3)\cap(XY-1,X^2-Y^3).$$ Now note that $(Y^2,X^2-Y^3)=(X^2,Y^2)$.

On the other side, the ideal $J=(XY-1,X^2-Y^3)$ is radical since $$\mathbb C[X,Y]/(XY-1,X^2-Y^3)\simeq\mathbb C[X,X^{-1}]/(X^5-1)$$ is a reduced ring. Then $J$ is the intersection of its minimal primes. If $P$ is a minimal prime over $J$, then $XY-1\in P$ and $X^2-Y^3\in P$. Then $X^5-1\in P$. Let $\omega$ be a primitive $5$th root of unity. Since $X^5-1=\prod_{i=0}^4(X-\omega^i)\in P$ we have that $X-\omega^i\in P$ for some $i$.
If $X-1\in P$, then $P=(XY-1,X-1)=(X-1,Y-1)$.
If $X-\omega\in P$ then $P=(XY-1,X-\omega)$, and so on.

This shows that $J=\bigcap_{i=0}^4(XY-1,X-\omega^i)$, and $I=(X^2,Y^2)\cap J$ is a primary decomposition.

Remark. From the foregoing we get the decomposition provided by Macaulay2: $I=(X^2,Y^2)\cap(X-1,Y-1)\cap(XY-1,X^4+X^3+X^2+X+1)$, but this is not a primary decomposition since the last ideal is not primary. (I think Macaulay2 considered $X^4+X^3+X^2+X+1$ irreducible.)

1. Using the primary decomposition of $I$ we get $$\mathbb C[X,Y]/I\simeq\mathbb C[X,Y]/(X^2,Y^2)\oplus\left(\oplus_{i=0}^4\mathbb C[X,Y]/(XY-1,X-\omega^i)\right).$$ The direct sum (for $i=0,\dots,4$) has length $5$, and $\mathbb C[X,Y]/(X^2,Y^2)$ has length $4$.