A prime chain in R[x] has length at most one more than its contraction chain

Let $\mathfrak{q}=P\cap R$. Localizing the given chain at $\mathfrak{q}$ preserves its length, so it is enough to bound the height of the prime $P_{\mathfrak{q}}\subset R_{\mathfrak{q}}[x]$. By [L1], after relabelling $\mathrm{ht}(\mathfrak{q})=r$ and choosing $a_1,\dots ,a_r\in \mathfrak{q}$, the maximal ideal $\mathfrak{q}{R}_{\mathfrak{q}}$ is minimal over $J=(a_1\mathrm{/}1,\dots ,a_r\mathrm{/}1)$.

If $P_{\mathfrak{q}}=\mathfrak{q}{R}_{\mathfrak{q}}[x]$, then $P_{\mathfrak{q}}$ is minimal over $J{R}_{\mathfrak{q}}[x]$: any prime of $R_{\mathfrak{q}}[x]$ containing $J$ contracts to a prime of $R_{\mathfrak{q}}$ containing $\mathfrak{q}{R}_{\mathfrak{q}}$, hence to $\mathfrak{q}{R}_{\mathfrak{q}}$ itself. Therefore [L2] gives $\mathrm{ht}(P_{\mathfrak{q}})\le r$.

Suppose now that $P_{\mathfrak{q}}\mathrm{\ne }\mathfrak{q}{R}_{\mathfrak{q}}[x]$. Then $P_{\mathfrak{q}}\mathrm{/}\mathfrak{q}{R}_{\mathfrak{q}}[x]$ is a nonzero prime ideal of $\kappa (\mathfrak{q})[x]$, where $\kappa (\mathfrak{q})=\mathrm{Frac}(R\mathrm{/}\mathfrak{q})$. By [L3], choose a lift $f\in R_{\mathfrak{q}}[x]$ whose image generates that nonzero prime of $\kappa (\mathfrak{q})[x]$. Any prime $Q\subseteq P_{\mathfrak{q}}$ containing $J$ and $f$ has contraction containing $J$, hence equal to $\mathfrak{q}{R}_{\mathfrak{q}}$ by step 1.1; modulo $\mathfrak{q}{R}_{\mathfrak{q}}[x]$, the prime $Q\mathrm{/}\mathfrak{q}{R}_{\mathfrak{q}}[x]$ contains the generator of $P_{\mathfrak{q}}\mathrm{/}\mathfrak{q}{R}_{\mathfrak{q}}[x]$, so it equals that prime. Therefore $Q=P_{\mathfrak{q}}$, and $P_{\mathfrak{q}}$ is minimal over the ideal $(J,f)$ generated by $r+1$ elements.

By [L2], step 2.2 gives $\mathrm{ht}(P_{\mathfrak{q}})\le r+1=\mathrm{ht}(\mathfrak{q})+1\le \dim R+1$. Since the localized chain still has length $n$, we have $n\le \mathrm{ht}(P_{\mathfrak{q}})\le \dim R+1$. Steps 2.1 and 3.1 cover both cases.

Therefore every strict prime chain in $R[x]$ has length at most $\dim R+1$.