# Two questions about Cohen-Macaulay rings

The following questions seem basic, but I can't find them in the literature. I'm also unable to think of a counterexample.

Let $A$ be a local Cohen-Macaulay ring of dimension $d$.

1. Let $I$ be an ideal generated by $r$ elements. Is is true that the depth of $A/I$ is at least $d-r$?

2. Let $Q$ be a minimal prime of $A$. Is it true that $A/Q$ is also Cohen-Macaulay?

1) Even when $A$ is regular, one can always find an ideal $I$ with $3$ generators such that $A/I$ has depth $0$. This is due to a very nice result by Bruns, which says you can construct $3$-generated ideal with all kinds of homological patern. The details are explained in this answer.
2) Let $R=k[X^4,X^3Y,XY^3,Y^4]$. Then $R$ is a domain of dimension $2$ which is not Cohen-Macaulay. So one can write $R=S/Q$, where $R=k[a,b,c,d]$ and $Q$ is a prime ideal of height 2. Take $(f,g)$ to be a regular sequence in $Q$ and let $I=(f,g)$. Then $A=S/I$ is Cohen-Macaulay (being a complete intersection), but $Q$ is a minimal prime of $A$ and $R=S/Q$ is not CM.
• The $P$ in item 2 is really $Q$? Apr 15, 2010 at 19:12
• Thank you for your nice answers! I am a bit confused about your answer to Greg's question: should one instead take $N$ to be the second syzygy of $k=[y, t_1, \cdots, t_n]/(t_1, \cdots, t_n)$? Apr 15, 2010 at 20:10
• But $A$ is not local, or am I missing something? Jun 22, 2010 at 8:26