# The Stacks Project

## Tag 02QY

Lemma 41.11.3. Let $(S, \delta)$ be as in Situation 41.8.1. 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. $\square$

The code snippet corresponding to this tag is a part of the file chow.tex and is located in lines 3359–3365 (see updates for more information).

\begin{lemma}
\label{lemma-cycle-closed-coherent}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Let $X$ be locally of finite type over $S$.
Let $Z \subset X$ be a closed subscheme.
If $\dim_\delta(Z) \leq k$, then $[Z]_k = [{\mathcal O}_Z]_k$.
\end{lemma}

\begin{proof}
This is because in this case the multiplicities $m_{Z', Z}$ and
$m_{Z', \mathcal{O}_Z}$ agree by definition.
\end{proof}

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