MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This is related to the question I asked last time. This sounds a bit too specific, I hope this question is still acceptable on MO.

I am still not quite comfortable with the concept of depth, and there is this exercise in Matsumura's book that goes as follows:

Find an example of a noetherian local ring $A$ and a finite $A$-module $M$ such that $\rm{depth}M > \rm{depth}A$. Also find $A$,$M$ and $P \in \rm{Spec}A$ such that $\rm{depth} M_P > \rm{depth}_P(M)$.

I hope I have found correct examples, but I am still quite lost about why one can find such examples, and what the generic ones are. So if someone can just give me some representative examples I would be grateful.

The examples I found myself:

For the first one, it is clear that $A$ must not be Cohen-Macaulay. Then I set $A = \frac {k[x,y,z]}{(xz,yz)}_{(x,y,z)}$, which is of depth 1, and I consider its quotient by $(z)$, which is $k[x,y]_{(x,y)}$ and should be of depth 2 (at least $x,y$ is a regular sequence I think).

For the second one, I try to fix $depth_P(M) = 0$, which means $P$ should lie in some associated primes of $M$, so I consider $M = \frac {k[x,y,z]}{(x^2,xy,xz)}_{(x,y)}$, such that $(x,y)$ is not associated prime when localized.

share|cite|improve this question
up vote 8 down vote accepted

1) Start with a regular local ring $R$. Take 2 ideals $I,J$ such that $I$ does not contain $J$, $R/I$ is CM and $\dim R/J <\dim R/I$. Then $A=R/(I\cap J)$ and $M=R/I$ work. In your example, $I=(x)$ and $J=(y,z)$. The reason is that CM means unmixed, so by having components of different dimensions one makes sure A is not CM.

2) Take $(A,m,k)$ to be any CM rings of dimension at least 2. Let $M=A\oplus k$. Then for any non-minimal $P\in Spec(A)-{m}$, $depth M_P =depth A_P$, but $depth M=0$.

The common theme: depth is usually the minimum depth of all components, while dimension is the maximal of those.

share|cite|improve this answer
nice examples. Thanks! – Ho Chung Siu Dec 3 '09 at 5:33

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.