# Maximal Cohen-Macaulay modules of full support, over non-artinian local Cohen-Macaulay rings, are faithful?

Let $$(R,\mathfrak m)$$ be a local Cohen-Macaulay ring of positive dimension. Let $$M$$ be a finitely generated maximal Cohen-Macaulay module i.e. $$\operatorname{depth} M=\dim R$$. If $$\operatorname{Supp}(M)=\operatorname{Spec}(R)$$, then is it true that $$M$$ is faithful?

My thoughts: Since $$\operatorname{Supp}(M)=\operatorname{Spec}(R)$$, so the radical of the annihilator ann$$_R(M)$$ is exactly the nilradical of $$R$$, so every element in the annihilator of $$M$$ is nilpotent. We would be done if $$R$$ were reduced, but that's not necessarily the case.

Let $$R=K[X,Y]/(X^2Y)$$ and $$M=R/(xy)$$. Then $$M$$ is MCM, $$\operatorname{Ann}_R(M)=(xy)$$, and $$\operatorname{Supp}(M)=\operatorname{Spec}(R)$$ since every prime ideal of $$R$$ contains $$x$$ or $$y$$.
• I just noticed that you want $R$ local. Then localize at $(x,y)$ or replace $K[X,Y]$ by $K[[X,Y]]$. Commented Aug 29, 2021 at 20:22