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 $k[x\_1,\dots,x\_d]\_{\mathfrak{m}^c}$?



Noether normalization with auxiliary conditions?
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.


