Some examples of depth

Question

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 $\operatorname{depth}M > \operatorname{depth}A$. Also find $A$, $M$ and $P \in \operatorname{Spec}A$ such that $\operatorname{depth}M_P > \operatorname{depth}_P(M)$.

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.

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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 $\operatorname{depth}_P(M) = 0$, 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.

Answer 1

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.