For a commutative ring $S$ of finite Krull dimension $d$, we have $1+d\leq \dim(S[X])\leq 2d+1$. One proof of this uses the fact that if $Q_1\subset Q_2\subset Q_3$ is a chain of prime ideals of $S[X]$, then $Q_1\cap S\subset Q_3\cap S$.

Now let $R$ be a commutative ring of infinite dimension. Let $\mathcal{C}$ be a chain of prime ideals in $R[X]$ of arbitrary cardinality. Consider $\mathcal{C_R}=\{Q\cap R: Q\in \mathcal{C}\}$, a chain of prime ideals in $R$. It follows from the fact in the previous paragraph that $\mathcal{C_R}$ is countably infinite if and only if $\mathcal{C}$ is countably infinite. Is it true in general that $\mathcal{C}$ and $\mathcal{C_R}$ have the same cardinality?