The Stacks project

Definition 82.15.1. In Situation 82.2.1 let $X/B$ be good. Let $k \in \mathbf{Z}$.

1. Given any locally finite collection $\{ W_ j \subset X\} $ of integral closed subspaces 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 $X$.