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 $k \in \mathbf{Z}$.

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.

We say that $\alpha \in Z_ k(X)$ is

*rationally equivalent to zero*if $\alpha $ is a cycle of the form displayed above.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.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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