Definition 42.19.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $k \in \mathbf{Z}$.

1. Given any locally finite collection $\{ W_ j \subset X\}$ of integral closed subschemes with $\dim _\delta (W_ j) = k + 1$, and any $f_ j \in R(W_ j)^*$ we may consider

$\sum (i_ j)_*\text{div}(f_ j) \in Z_ k(X)$

where $i_ j : W_ j \to X$ is the inclusion morphism. This makes sense as the morphism $\coprod i_ j : \coprod W_ j \to X$ is proper.

2. We say that $\alpha \in Z_ k(X)$ is rationally equivalent to zero if $\alpha$ is a cycle of the form displayed above.

3. We say $\alpha , \beta \in Z_ k(X)$ are rationally equivalent and we write $\alpha \sim _{rat} \beta$ if $\alpha - \beta$ is rationally equivalent to zero.

4. We define

$\mathop{\mathrm{CH}}\nolimits _ k(X) = Z_ k(X) / \sim _{rat}$

to be the Chow group of $k$-cycles on $X$. This is sometimes called the Chow group of $k$-cycles modulo rational equivalence on $X$.

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