Let $R$ be a regular, local $\mathbb{Q}$-algebra with a regular system of parameters $x\_1, \dotsc, x\_n$, and let 
$$f \colon \mathbb{Q}[X\_1, \dotsc, X\_n]\_{(X\_1, \dotsc, X\_n)} \rightarrow R$$
be the map given by $X\_i \mapsto x\_i$. Then $f$ is flat (for instance, by Bourbaki, cf. EGA III, 0.10.2.2). 


Is $f$ a regular morphism, that is, are the (geometric) fibers of $f$ regular? Certainly, the closed fiber is regular, but how about the others?


The answer should be yes, and I would appreciate an argument for this. In the case when $R$ is excellent, the positive answer seems to be a special case of EGA IV, 7.9.8, but a less contrived argument (and one that would not use an additional excellence assumption) would be greatly appreciated.



Is this morphism of regular local rings regular?
You do not need $R$ excellent. Since $\mathbb{Q}[X\_1, \dotsc, X\_n]\_{(X\_1, \dotsc, X\_n)}$ is excellent it follows from [M. André, Localisation de la lissité formelle.](https://link.springer.com/article/10.1007/BF01168230)


