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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