Lemma 62.8.4 (0H5U)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 62: Relative Cycles
Section 62.8: Effective relative cycles
Lemma 62.8.4 (cite)

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Lemma 62.8.4. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r \geq 0$ be an integer. Let $\alpha$ be a relative $r$ -cycle on $X/S$ . Let $g : S' \to S$ be a surjective morphism. Then $\alpha$ is effective if and only if the base change $g^*\alpha$ is effective.

Proof. Omitted. $\square$

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