Lemma 62.7.3. Let $f : X \to S$ be a morphism of schemes. Assume $S$ is locally Noetherian and $f$ is locally of finite type. Let $r \geq 0$ be an integer. Let $\alpha $ be a relative $r$-cycle on $X/S$. If $\alpha $ is equidimensional, then any restriction, base change, or flat pullback of $\alpha $ is equidimensional.

**Proof.**
Omitted.
$\square$

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