Lemma 62.7.6. 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 $X/S$. Let $\{ f_ i : X_ i \to X\} $ be a jointly surjective family of flat morphisms, locally of finite type, and of relative dimension $e$. Then $\alpha $ is equidimensional if and only if each flat pullback $f_ i^*\alpha $ is equidimensional.
Proof. Omitted. Hint: As in the proof of Lemma 62.7.5 one shows that the inverse image by $f_ i$ of the closure $W$ of the support of $\alpha $ is the closure $W_ i$ of the support of $f_ i^*\alpha $. Then $W \to S$ has relative dimension $\leq r$ holds if $W_ i \to S$ has relative dimension $\leq r + e$ for all $i$. $\square$
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