Is it true that if $S/\mathbb{Z}_p$ is a flat extension of discrete valuation rings then $S$ is finite over $\mathbb{Z}_p$?
2026-03-25 20:19:09.1774469949
Is a flat extension of DVRs always finite?
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No. Consider $\mathbb{Z}_p \subset \mathcal{O}_{\mathbb{Q}_p^{nr}}$ or $\mathbb{Q}[[t]] \subset \mathbb{C}[[t]]$.