Lemma 62.9.3. 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_ i : S_ i \to S\} $ be a h covering. Then $\alpha $ is proper if and only if each base change $g_ i^*\alpha $ is proper.
Proof. If $\alpha $ is proper, then each $g_ i^*\alpha $ is too by Lemma 62.9.2. Assume each $g_ i^*\alpha $ is proper. To prove that $\alpha $ is proper, it clearly suffices to work affine locally on $S$. Thus we may and do assume that $S$ is affine. Then we can refine our covering $\{ S_ i \to S\} $ by a family $\{ T_ j \to S\} $ where $g : T \to S$ is a proper surjective morphism and $T = \bigcup T_ j$ is an open covering. It follows that $\beta = g^*\alpha $ is proper on $Y = T \times _ S X$ over $T$. By Lemma 62.5.7 we find that the support of $\beta $ is the inverse image of the support of $\alpha $ by the morphism $f : Y \to X$. Hence the closure $W \subset Y$ of $f^{-1}\text{Supp}(\alpha )$ is proper over $T$. Since the morphism $T \to S$ is proper, it follows that $W$ is proper over $S$. Then by Cohomology of Schemes, Lemma 30.26.5 the image $f(W) \subset X$ is a closed subset proper over $S$. Since $f(W)$ contains $\text{Supp}(\alpha )$ we conclude $\alpha $ is proper. $\square$
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