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Let $X$ be a $R$-scheme, where $R$ is a dvr. Suppose that the reduction of $X$ (over the closed point of $\mathrm{Spec} \ R$) is smooth and projective. Does this imply that $X$ is smooth and projective? Smoothness might get messed up by the generic fibre, but what about projectiveness?

Is $X\to \mathrm{Spec} R$ projective?

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up vote 2 down vote accepted

No. Consider a finite unramified extension $R'/R$ such that $R'$ is integral with two maximal ideals. Let $X$ be $\mathrm{Spec}(R')$ minus one closed point. Then $X$ is quasi-finite, étale, but not finite (hence not projective) over $R$, and its closed fiber is smooth and finite.

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