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I am starting to study Cohen-Macaulay rings, mainly from Bruns-Herzog book.

In that book there are many examples and sentences of the type "If something satisfies this properties, then it is Cohen-Macaulay".

The definition of Cohen-Macaulay (depth=dim) is clear to me. What I am missing at the moment are the implications arising as converse of the previous. Accidentally this implications could also answer the question "Why do we study Cohen Macaulay rings, apart from the fact that they are nice?"

So, what I am basically asking is to substitute something nice to X in the following sentence (although not necessarily simple to understand):

"If something is Cohen-Macaulay then X"

Anyone who wants to try and help me?

Thanks in advance,

Davide

user26857
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dadexix86
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  • I think almost all rings one encounter in number theory research nowadays are Cohen-Macaulay. – Bombyx mori Nov 19 '13 at 21:37
  • Please be specific about the examples. You don't want converses of the unspecified statements, or you'd specify the statements, right? – Thomas Andrews Nov 19 '13 at 21:37
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    @DanielFischer Not wise to try to make culturally witty comments when the OP is clearly not comfortable with English... – Thomas Andrews Nov 19 '13 at 21:38
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    @ThomasAndrews I absolutely thought that was an intentional allusion by the OP. – Daniel Fischer Nov 19 '13 at 21:42
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    @ThomasAndrews, DanielFischer is right. I intentionally alluded to that famous title by Oscar Wilde. ;)

    By the way, I am sorry for my bad English.

    Going back to my question, what I am asking is not the converse of some specific question, so that it turns out to be a "if and only if".

    I am looking for (optionally nice) proposition/properties in which being Cohen-Macaulay is the hypothesis and not the thesis.

     – dadexix86 Nov 19 '13 at 21:47
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    For one thing, if $A$ is C-M then $\mathrm{Spec } A$ is equidimensional (all irreducible components have the same dimension). – Fredrik Meyer Nov 20 '13 at 10:46
  • @FredrikMeyer well, this is a nice property indeed! Thanks :) – dadexix86 Nov 20 '13 at 10:54
  • PS: Can you provide reference for that? – dadexix86 Nov 20 '13 at 11:06
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    This is Corollary 18.11 in Eisenbud's book, page 458 – Fredrik Meyer Nov 20 '13 at 11:45

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