Remark 62.5.6 (Support). Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $r \geq 0$ be an integer. Let $\alpha $ be a family of $r$-cycles on fibres of $X/S$. We define the support of $\alpha $ to be
\[ \text{Supp}(\alpha ) = \bigcup \nolimits _{s \in S} \text{Supp}(\alpha _ s) \subset X \]
Here $\text{Supp}(\alpha _ s) \subset X_ s$ is the support of the cycle $\alpha _ s$, see Chow Homology, Definition 42.8.3. The support $\text{Supp}(\alpha )$ is rarely a closed subset of $X$.
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