Actually I was thinking of a finite (possibly ramified) extension $K$ of $\mathbb Q_p$ and the rings of integers $\mathbb Z_p$ an $O_K$. Is $\text{Spec } O_K \to \text{Spec }\mathbb Z_p $ smooth?
Is any extension of DVRs smooth?
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abstract-algebra
algebraic-geometry
ring-theory
1 Answers
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I'm going to answer this hoping that someone provides a better (or at least more general) answer. I've been thinking about this question myself and was surprised I couldn't find this topic discussed anywhere I looked.
I don't think $\mathbb{Z}_2 \rightarrow \mathbb{Z}_2[\sqrt{2}]$ is formally smooth (hence it's not smooth). Consider the map $(\mathbb{Z}/4\mathbb{Z})[x]/(x^2) \rightarrow (\mathbb{Z}/2\mathbb{Z})[x]/(x^2)$; this corresponds to quotienting by the ideal $(2)$, which is a square zero ideal. The map $\mathbb{Z}_2[\sqrt{2}] \rightarrow (\mathbb{Z}/2\mathbb{Z})[x]/(x^2)$ given by $\sqrt{2} \mapsto x$ does not lift to a map $\mathbb{Z}_2[\sqrt{2}] \rightarrow (\mathbb{Z}/4\mathbb{Z})[x]/(x^2)$, so $\mathbb{Z}_2[\sqrt{2}]$ is not formally smooth over $\mathbb{Z}_2$.