# An example of a local ring which is not CM and a MCM module over it

I am looking for an example of a commutative noetherian local ring $$(A,m)$$, and a maximal Cohen-Macaulay module $$M$$ over $$A$$ (in particular $$M$$ is finitely generated over $$A$$), such that for some $$p \in Spec(A)$$, we have that $$M_p = 0$$, and $$A_p$$ is also not a Cohen-Macaulay ring. Do there exist such examples?

• Let $A$ be the local ring of a union of two $2$-planes intersecting in one point $x$ in a $4$-dimensional ambient space. Let $M$ be the local ring of one of the two irreducible components. Let $p$ be a prime ideal of a curve in the other component that contains $x$. If you really want $A$ to be not Cohen-Macaulay at $p$, then just add some embedded structure to $A$ along $p$. Nov 29 '18 at 14:16

Let $$A=k[x,y,z]_{(x,y,z)} / (x^2z, xyz)$$. Let $$p=(x,y)A$$. Let $$M=A/zA$$, thought of as a cyclic $$A$$-module. Then $$M \cong k[x,y]_{(x,y)}$$ is a maximal Cohen-Macaulay module over $$A$$, since $$\dim A = \operatorname{depth} M = 2$$. We have that $$p$$ is prime because $$A/p \cong k[z]_{(z)}$$. We have that $$M_p = 0$$ because $$z \notin p$$, so $$z$$ acts like a unit on $$A_p$$-modules but kills $$M$$. Finally, $$A_p \cong k(z)[x,y]_{(x,y)} / (x^2, xy)$$ has dimension 1 and depth 0.