Some examples of depth - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T10:36:51Z http://mathoverflow.net/feeds/question/7556 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7556/some-examples-of-depth Some examples of depth Ho Chung Siu 2009-12-02T05:11:34Z 2009-12-02T19:20:57Z <p>This is related to the <a href="http://mathoverflow.net/questions/6704/how-to-think-about-cm-rings" rel="nofollow">question</a> I asked last time. This sounds a bit too specific, I hope this question is still acceptable on MO.</p> <p>I am still not quite comfortable with the concept of depth, and there is this exercise in Matsumura's book that goes as follows:</p> <blockquote> <p>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 $\rm{depth} M_P > \rm{depth}_P(M)$. </p> </blockquote> <p>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.</p> <p><hr /></p> <p>The examples I found myself:</p> <p>For the first one, it is clear that $A$ must not be Cohen-Macaulay. Then I set <code>$A = \frac {k[x,y,z]}{(xz,yz)}_{(x,y,z)}$</code>, 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).</p> <p>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 <code>$M = \frac {k[x,y,z]}{(x^2,xy,xz)}_{(x,y)}$</code>, such that $(x,y)$ is not associated prime when localized.</p> http://mathoverflow.net/questions/7556/some-examples-of-depth/7564#7564 Answer by Hailong Dao for Some examples of depth Hailong Dao 2009-12-02T06:32:23Z 2009-12-02T19:20:57Z <p>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 $\dim R/J &lt;\dim R/I$. 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. </p> <p>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 $depth M=0$.</p> <p>The common theme: depth is usually the minimum depth of all components, while dimension is the maximal of those. </p>