Recall that a discrete valuation ring $R$ is *excellent* if the extension $\widehat{K}/K$ is separable, where $\widehat{R}$ is the completion of $R$ (with respect to the maximal ideal), $K = \mathrm{Frac}(R)$, and $\widehat{K} = \mathrm{Frac}(\widehat{R})$. Suppose therefore that $R$ is excellent and consider a local injection $R \hookrightarrow R^{\prime}$ of discrete valuation rings such that the induced residue field extension is separable and such that a uniformizer of $R$ is also a uniformizer of $R^{\prime}$. Then, is $R^{\prime}$ necessarily excellent?



"Unramified" extension of DVRs and permanence of excellence
No. Let $R'$ be any non-excellent dvr whatsoever, hence of equicharacteristic $p > 0$, and let $t \in R'$ be a uniformizer. Let $R = \mathbf{F}\_p[t]\_{(t)}$. The local inclusion $R \hookrightarrow R'$ has induced residue field extension that is separable since $\mathbf{F}\_p$ is perfect. And of course $R$ is excellent for any of a million reasons. So this seems to be a counterexample.


