polynomial local regularity fibre step

Localize first at $R\setminus \mathfrak{p}$, quotient by $\mathfrak{p}{R}_{\mathfrak{p}}$, and then localize at the image of $\mathfrak{q}$. Fractions and the quotient relation identify the fibre with $k(\mathfrak{p})[t{]}_{\bar{\mathfrak{q}}}$. Over this field the polynomial ring is Euclidean and hence a PID.

In the PID $k(\mathfrak{p})[t]$, every nonzero prime is generated by an irreducible polynomial $f$ and is maximal. Localization at it is a nonfield local PID with maximal ideal $(f)$ and dimension one. Since $f\mathrm{\notin }(f^2)$ by cancellation, its cotangent space $(f)\mathrm{/}(f^2)$ is one-dimensional over the residue field, so the localization is regular by [F1]; it is a DVR by the powers-of-$f$ valuation. The zero-prime localization is the rational function field, which has zero maximal ideal and dimension zero and is regular by [F1].