Lemma 62.8.7. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r, e \geq 0$ be integers. Let $\alpha $ be a relative $r$-cycle on $X/S$. If $\alpha $ is effective, then $\alpha $ is equidimensional.

**Proof.**
Assume $\alpha $ is effective. By Lemma 62.8.6 the support $\text{Supp}(\alpha )$ is closed in $X$. Thus $\alpha $ is equidimensional as the fibres of $\text{Supp}(\alpha ) \to S$ are the supports of the cycles $\alpha _ s$ and hence have dimension $r$.
$\square$

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