Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

$A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. Then is it true that $\mbox{depth }M\le\mbox{depth }A$ just like $\mbox{dim }M\le\mbox{dim }A$? I don't see any relation between an $M$-sequence and an $A$-sequence. At least I know it is true when $\mbox{inj.dim }M<\infty$, from the relation $\mbox{depth }M\leq\mbox{dim }M\leq\mbox{inj. dim }M=\mbox{depth }A\leq\mbox{dim }A$. But what happens when $\mbox{inj.dim }M=\infty$? Another inequality I'm not quite sure about when $\mbox{inj.dim }M=\infty\ $: is it true that $\mbox{dim }M\leq\mbox{depth }A$?

share|improve this question
1  
Noetherian, both here and in the other question. –  Victor Protsak Aug 6 '10 at 4:51
1  
Auslander-Buchsbaum's theorem says when $\mbox{proj.dim }M<\infty$ $\mbox{depth }A-\mbox{depth }M=\mbox{proj.dim }M$ so it appears the inequality holds when either projective dimension or injective dimension is finite. –  ashpool Aug 6 '10 at 13:38
    
I retagged, since this was on the front page anyway –  David White Jan 11 '12 at 18:48

2 Answers 2

up vote 11 down vote accepted

$A=k[[x,y]]/(x^2,xy)$ then depth$(A)=0$. Let $M=R/(x)=k[[y]]$ then $y$ is a nonzerodivisor on $M$.

share|improve this answer
    
What a remarkable example. Thanks! –  ashpool Aug 6 '10 at 14:18
    
Did you mean $M=A/(x)$? –  Sándor Kovács Jan 11 '12 at 16:28
    
yes, thanks Sandor. –  Manish Kumar Aug 13 '12 at 12:13

In the paper "Eine Dualität zwischen den Funktoren Ext und Tor" (J. Algebra 11, 510–531) Ischebeck shows that if $A$ admits a finitely generated module $N$ of finite injective dimension, then the answer is affirmative. More precisely, for any finitely generated module $M$ one has $\text{depth}\ A - \text{depth}\ M = \sup\left\lbrace i : \text{Ext}^i_A(M,N) \neq 0 \right\rbrace $. This is Excercise 3.1.24 in Bruns/Herzog "Cohen-Macaulay-Rings". In that chaper there is more material on rings that admit a finitely generated module of finite injective dimension.

share|improve this answer

Your Answer

 
discard

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.