## 81.15 Rational equivalence

This section is the analogue of Chow Homology, Section 42.19. In this section we define *rational equivalence* on $k$-cycles. We will allow locally finite sums of images of principal divisors (under closed immersions). This leads to some pretty strange phenomena (see examples in the chapter on schemes). However, if we do not allow these then we do not know how to prove that capping with Chern classes of line bundles factors through rational equivalence.

Definition 81.15.1. In Situation 81.2.1 let $X/B$ be good. Let $k \in \mathbf{Z}$.

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.

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 $X$*.

There are many other interesting equivalence relations. Rational equivalence is the coarsest of them all. A very simple but important lemma is the following.

Lemma 81.15.2. In Situation 81.2.1 let $X/B$ be good. Let $U \subset X$ be an open subspace. Let $Y$ be the reduced closed subspace of $X$ with $|Y| = |X| \setminus |U|$ and denote $i : Y \to X$ the inclusion morphism. Let $k \in \mathbf{Z}$. Suppose $\alpha , \beta \in Z_ k(X)$. If $\alpha |_ U \sim _{rat} \beta |_ U$ then there exist a cycle $\gamma \in Z_ k(Y)$ such that

\[ \alpha \sim _{rat} \beta + i_*\gamma . \]

In other words, the sequence

\[ \xymatrix{ \mathop{\mathrm{CH}}\nolimits _ k(Y) \ar[r]^{i_*} & \mathop{\mathrm{CH}}\nolimits _ k(X) \ar[r]^{j^*} & \mathop{\mathrm{CH}}\nolimits _ k(U) \ar[r] & 0 } \]

is an exact complex of abelian groups.

**Proof.**
Let $\{ W_ j\} _{j \in J}$ be a locally finite collection of integral closed subspaces of $U$ of $\delta $-dimension $k + 1$, and let $f_ j \in R(W_ j)^*$ be elements such that $(\alpha - \beta )|_ U = \sum (i_ j)_*\text{div}(f_ j)$ as in the definition. Let $W_ j' \subset X$ be the corresponding integral closed subspace of $X$, i.e., having the same generic point as $W_ j$. Suppose that $V \subset X$ is a quasi-compact open. Then also $V \cap U$ is quasi-compact open in $U$ as $V$ is Noetherian. Hence the set $\{ j \in J \mid W_ j \cap V \not= \emptyset \} = \{ j \in J \mid W'_ j \cap V \not= \emptyset \} $ is finite since $\{ W_ j\} $ is locally finite. In other words we see that $\{ W'_ j\} $ is also locally finite. Since $R(W_ j) = R(W'_ j)$ we see that

\[ \alpha - \beta - \sum (i'_ j)_*\text{div}(f_ j) \]

is a cycle on $X$ whose restriction to $U$ is zero. The lemma follows by applying Lemma 81.10.2.
$\square$

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