Definition 62.9.1. Let $f : X \to S$ be a morphism of schemes. Assume $S$ is locally Noetherian and $f$ is locally of finite type. Let $r \geq 0$ be an integer. We say a relative $r$-cycle $\alpha $ on $X/S$ is a proper relative cycle if the support of $\alpha $ (Remark 62.5.6) is contained in a closed subset $W \subset X$ proper over $S$ (Cohomology of Schemes, Definition 30.26.2). The group of all proper relative $r$-cycles on $X/S$ is denoted $c(X/S, r)$.
62.9 Proper relative cycles
In our setting, the following is probably the correct definition.
By Cohomology of Schemes, Lemma 30.26.3 this just means that the closure of the support is proper over the base. To see that these form a group, use Cohomology of Schemes, Lemma 30.26.6.
Lemma 62.9.2. Let $f : X \to S$ be a morphism of schemes. Assume $S$ is locally Noetherian and $f$ is locally of finite type. Let $r \geq 0$ be an integer. Let $\alpha $ be a relative $r$-cycle on $X/S$. If $\alpha $ is proper, then any base change $\alpha $ is proper.
Proof. Omitted. $\square$
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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