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