The Stacks project

Lemma 33.19.2. Let $f : X \to \mathop{\mathrm{Spec}}(R)$ be a morphism from an irreducible scheme to the spectrum of a valuation ring. If $f$ is locally of finite type and surjective, then the special fibre is equidimensional of dimension equal to the dimension of the generic fibre.

Proof. We may replace $X$ by its reduction because this does not change the dimension of $X$ or of the special fibre. Then $X$ is integral and the lemma follows from Algebra, Lemma 10.125.9. $\square$

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