Lemma 42.10.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme. If $\dim _\delta (Z) \leq k$, then $[Z]_ k = [{\mathcal O}_ Z]_ k$.

**Proof.**
This is because in this case the multiplicities $m_{Z', Z}$ and $m_{Z', \mathcal{O}_ Z}$ agree by definition.
$\square$

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