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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?

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up vote 17 down vote accepted

Nice questions! The answers are no in both cases, although the examples are more interesting than one would expect.

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.

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The $P$ in item 2 is really $Q$? – Steven Sam Apr 15 '10 at 19:12
@Steven: that's right, thanks! – Hailong Dao Apr 15 '10 at 19:15
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 $k=[y, t_1, \cdots, t_n]/(t_1, \cdots, t_n)$? – user1594 Apr 15 '10 at 20:10
@JT: That's a good point. Actually the statement of Bruns's theorem requires N to be 2nd syzygy of a module of depth at least 1, so we can't take 2nd syzygy of k. – Hailong Dao Apr 15 '10 at 21:46
I now clarify the point above in my answer there as well. – Hailong Dao Apr 15 '10 at 21:59

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