regular locus of an affine domain

Let $A$ be an affine domain over a field $k$ (need not be algebraically closed). Let $\mathfrak{p}$ be a prime ideal of $A$, such that $A_{\mathfrak{p}}$ is a regular local ring. Does there always exist a maximal ideal $\mathfrak{m}$ of $A$, such that $\mathfrak{p}\subseteq \mathfrak{m}$, and $A_{\mathfrak{m}}$ is also a regular local ring?

• Don't understand why people are in such a hurry to reply without even understanding the question! – sagnik chakraborty Mar 6 '15 at 10:05
• How about taking the localization of $k[X,Y]/(Y²-X²(X+1))$ at the nodal singularity? The result is local ring of dimension 1 which is not regular. However, the field of fractions is a regular local ring. This should be a negative answer to the question. – Matthias Wendt Mar 6 '15 at 10:15
• I need some maximal ideal (not all!) containing the prime ideal. Sorry to say, but you've also failed to understand the question! – sagnik chakraborty Mar 6 '15 at 10:24
• @MatthiasWendt No need to assume characteristic 0. What you say works for any affine algebra over any field. – ulrich Mar 6 '15 at 11:44
• The regular locus is open more generally for finite type algebras over a complete local ring (search for "excellent rings" if you want to explore this condition in more generality, for instance in EGA IV, 6 and 7). – Vinteuil Mar 6 '15 at 12:47