# Why is this ring not Cohen-Macaulay?

I am stuck with exercise 18.8, page 466 of Eisenbud's Commutative Algebra with a view towards Algebraic Geometry:

Let $k$ be a field. The task is to prove that $R:=k[x^4, x^3y, xy^3, y^4] \subseteq k[x,y]$ is not a Cohen-Macaulay ring, i.e. the length of a maximal regular sequence (i.e. a sequence of elements $a_1, \dots , a_r$ such that $a_i$ is not a zero divisor in $R/(a_1, \dots , a_{i-1})$) is smaller than the Krull dimension (length of a maximal chain of prime ideals).

The Krull dimension of the ring is bigger or equal than 2, hence it is enough to show that the length of a maximal regular sequence is at most one.

Our ring has the maximal ideal $m = (x^4, x^3y, xy^3, y^4)$ and for this we have the "system of parameters" $\{x^4, y^4\}$ (where "system of parameters" means that there exists an $n$ such that $m^n \subseteq (x^4, y^4)$ --- here $n=4$ works). There is a theorem saying that, given such a system of parameters, a maximal regular sequence can be built out of its elements (and any two maximal regular sequences have the same length). So it is enough to look how long a regular sequence we can build out of $\{x^4, y^4\}$. Since we want to show that we cannot achieve a sequence of length two, it is enough to show that $(x^4, y^4)$ is not a regular sequence.

This amounts to showing that $y$ is a zero divisor $R/(x^4)$. This should be a very concrete calculation but it is where I am stuck. Any help would be appreciated!

Certainly $(x^3y)^2y^4 = x^6y^6 = x^4(xy^3)^2$ is zero when you mod out by $x^4$. The key thing to see is that $(x^3y)^2$ is not inside $(x^4)$. Why is this?
• Ah, thanks! This is the right question to ask and which I couldn't come up with! In $k[x,y]$ we have $(x^3y)^2=x^4(x^2y^2)$. Since $k[x,y]$ is an integral domain, hence cancellative as a multiplicative monoid, $x^2y^2$ is the only element we can multiply with $x^4$ to get $(x^3y)^2$. Hence it is enough to show that $x^2y^2$ is not in the subring $R$. But this is true because no monomial in $R$ has degree in $x$ and in $y$ both smaller than $3$. – Who Oct 7 '13 at 19:40