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This is from CM Rings (Bruns & Herzog) p 64 2.1.20. Show that a one dimensional Noetherian local ring has a maximal Cohen Maucaulay module.

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closed as too localized by Mark Sapir, Hailong Dao, Dan Petersen, Bill Johnson, Andy Putman Jan 31 '12 at 23:52

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up vote 3 down vote accepted

Take $R/\mathfrak{p}$, where $\mathfrak{p}$ is a minimal prime ideal of $R$.

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In this case R/p has dimension 1, but why it has depth 1? R is not necessarily CM, so R can have depth 0. – Andrei Jan 31 '12 at 9:37
$R/p$ is a domain, so has depth 1, both as a ring itself and as a module over the original ring. – Graham Leuschke Jan 31 '12 at 11:43

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