Example of a ring which is not CM at all its prime ideals

Question

A commutative ring $A$ is said to be CM at a maximal ideal $\mathfrak{m}$ if and only if $Depth(A_{\mathfrak{m}})=Krull(A_{\mathfrak{m}})$.
What is an example of a connected commutative ring $A$ which is CM at one of its ideals but not all of them?

Answer 1

Consider the ring $A = k[x,y]/(x^2, xy) = k[x,y]/\left((x)\cap(x,y)^2\right)$. Geometrically this is the line $x=0$ in the plane, with an embedded point at the origin. It is regular at every maximal ideal other than $\mathfrak m = (x,y)\subseteq A$, hence is CM away from this ideal a fortiori. However, at the origin, the depth is $0$. To see this, let $f/g\in \mathfrak m A_{\mathfrak m}$, so that $f = bx + c_1y + \cdots c_dy^d$, and note that $f/g$ is a zero-divisor since $x\cdot (f/g) = (xf)/g = 0$. (We see that $x = x/1\neq 0$ in the localization, since the converse holds iff there is some $u\notin \mathfrak m$ such that $ux = 0$.)

In this example, the ring is CM at every maximal ideal except for the one with the embedded point. This example is the simplest case of a more general phenomenon: a $1$-dimensional (locally noetherian) scheme is CM iff it has no embedded points.