Krull dimension of schemes locally of finite type over PID

Let $$R$$ be a commutative unital ring that is a PID. Assume that $$R$$ is not a DVR. Let $$X$$ be an integral scheme locally of finite type over $$\mathrm{Spec}\,R$$. Can the Krull dimension of $$\mathcal{O}_X(X)$$ exceed that of $$X$$? This is not true if we drop all assumptions on $$X$$.