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Let $k$ be a field and $R$ a $d$-dimensional integral domain over $k$. Noether normalization produces an injective algebra homomorphism $k[x_1,\dots, x_d] \to R$ which is module-finite.

Given a maximal ideal $\mathfrak{m} \in \mathrm{Spec}(R)$, can one always find a Noether normalization such that $R_{\mathfrak{m}}$ is flat (= free) over $k[x_1,\dots,x_d]_{\mathfrak{m}^c}$?

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No, this would force $R_\mathfrak{m}$ to be Cohen-Macaulay. So a non-CM domain like $R=k[X^4,X^3Y,XY^3,Y^4]$ would be a counterexample.

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    $\begingroup$ In fact, not being Cohen-Macaulay is the only obstruction. A Noether normalization will be flat at a maximal ideal $\mathcal m$ if and only if $R_{\mathfrak m}$ is Cohen-Macaulay. $\endgroup$
    – Dustin Cartwright
    Commented Jan 25, 2012 at 18:20

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