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$A$ 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 $\mbox{dim }M\leq\mbox{depth }A$?

edited Jan 11 2012 at 18:47

David White
5,050●1●20●27

asked Aug 5 2010 at 23:45

ashpool
1,419●4●16

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Noetherian, both here and in the other question. – Victor Protsak Aug 6 2010 at 4:51

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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 2010 at 13:38

I retagged, since this was on the front page anyway – David White Jan 11 2012 at 18:48

Manish Kumar · Accepted Answer · 2010-08-06 00:51:36Z UTC

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

answered Aug 6 2010 at 0:51

Manish Kumar
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	What a remarkable example. Thanks! – ashpool Aug 6 2010 at 14:18
	Did you mean $M=A/(x)$? – Sándor Kovács Jan 11 2012 at 16:28
	yes, thanks Sandor. – Manish Kumar Aug 13 at 12:13

Thomas Kahle · Answer 2 · 2012-01-11 15:35:56Z UTC

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.

Depth and dimension

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