The Stacks project

Example 62.5.4. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $r \geq 0$ be an integer. Let $Z \subset X$ be a closed subscheme. For $s \in S$ denote $Z_ s$ the inverse image of $Z$ in $X_ s$ or equivalently the scheme theoretic fibre of $Z$ at $s$ viewed as a closed subscheme of $X_ s$. Assume $\dim (Z_ s) \leq r$ for all $s \in S$. Then we can associate to $Z$ the family $[Z/X/S]_ r$ of $r$-cycles on fibres of $X/S$ defined by the formula

\[ [Z/X/S]_ r = ([Z_ s]_ r)_{s \in S} \]

where $[Z_ s]_ r$ is given by Chow Homology, Definition 42.9.2.

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