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The Stacks project

Lemma 62.4.2. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $\kappa $. Let $X$ be a scheme locally of finite type over $R$. Let $r \geq 0$. Let $W \subset X$ be a closed subscheme flat over $R$. Assume $\dim (W_ K) \leq r$. Then $\dim (W_\kappa ) \leq r$ and

\[ sp_{X/R}([W_ K]_ r) = [W_\kappa ]_ r \]

Proof. Taking $\mathcal{F} = \mathcal{O}_ W$ this is a special case of Lemma 62.4.1. See Chow Homology, Lemma 42.10.3. $\square$

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