Lemma 81.6.3. In Situation 81.2.1 let $X/B$ be good. Let $Y \subset X$ be a closed subspace. If $\dim _\delta (Y) \leq k$, then $[Y]_ k = [i_*\mathcal{O}_ Y]_ k$ where $i : Y \to X$ is the inclusion morphism.

Proof. Let $Z$ be an integral closed subspace of $X$ with $\dim _\delta (Z) = k$. If $Z \not\subset Y$ the $Z$ has coefficient zero in both $[Y]_ k$ and $[i_*\mathcal{O}_ Y]_ k$. If $Z \subset Y$, then the generic point of $Z$ may be viewed as a point $y \in |Y|$ whose image $x \in |X|$. Then the coefficient of $Z$ in $[Y]_ k$ is the length of $\mathcal{O}_ Y$ at $y$ and the coefficient of $Z$ in $[i_*\mathcal{O}_ Y]_ k$ is the length of $i_*\mathcal{O}_ Y$ at $x$. Thus the equality of the coefficients follows from Lemma 81.4.3. $\square$

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