The Stacks project

Lemma 62.8.5. 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 effective if and only if each flat pullback $f_ i^*\alpha $ is effective.

Proof. Omitted. $\square$

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