Let be $R=\mathbb{C} \lbrace x,y,z \rbrace$ the formal series ring and let $f_{1},f_{2},f_{3} \in R$ be nonzero elements of $R$.

(a) Consider the varieties $M:=V(f_{1},f_{2})$ and $N:=V(f_{2},f_{3})$ in $\mathbb{C}^{3}$ and suppose that $M$ and $N$ are curves with at least two irreducible components and exactly one irreducible component in common. Now, consider the intersection $Q=M \cap N$ given by $Q=V(f_{1},f_{2},f_{3})$, the local ring of $Q$ is:

$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathcal{O}_{Q}=\mathbb{C}\lbrace x,y,z \rbrace /(f_{1},f_{2},f_{3}) $

I think that $\mathcal{O}_{Q}$ is not Cohen-Macaulay, but I do not know how to prove.

$\mathcal{O}_{Q}$ is not Cohen-Macaulay?

(b) In general, consider two curves $M$ and $N$ in $\mathbb{C}^n$ with at least two irreducible components (that is, $M$ and $N$ are not irreducible) and exactly one irreducible component in common (which we call $Q$). Thus, the intersection of $M$ and $N$ is the curve $Q$.

The curve $Q$ is not Cohen Macaulay?