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I know this is a simple question but to make sure...:

$A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ we have $\dim A_{\mathfrak{m}}=\dim A$. Is then $A$ regular?

This seems wrong... why?

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Any reduced noetherian ring of dimension $1$ is Cohen-Macaulay, but they are not regular in general.

Example: $A=\mathbb C[x,y]/(y^2-x^3)$.

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For instance, there are local CM rings that aren't regular.

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  • $\begingroup$ I know but I was wondering when is a local CM ring regular? $\endgroup$
    – ABIM
    Aug 13, 2014 at 1:35
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    $\begingroup$ @CSA: Then perhaps that is the question that you should ask! $\endgroup$
    – RghtHndSd
    Aug 13, 2014 at 20:44

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