## 62.9 Proper relative cycles

In our setting, the following is probably the correct definition.

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 $c(X/S, r)$.

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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