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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 $X$.

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