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I'm reading Matsumura's Commutative ring theory and in chapter 11 he proves that if A is a noetehrian complete local k-algebra (k a field), then if A is m-smooth over k (m the maximal ideal of A), then A is a regular ring. Is it true that m-smoothness implies regularity when A is just a noetherian complete local ring and does not contain any field? thank you.

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m-smoothness over what? –  Graham Leuschke Dec 3 '10 at 2:48
m-smoothness over Z for example, or if A is an R-algebra over R. –  user11270 Dec 4 '10 at 4:01

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