The Stacks project

82.15 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 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 $X$.

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 82.15.2. In Situation 82.2.1 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 82.10.2. $\square$

Remark 82.15.3. In Situation 82.2.1 let $X/B$ be good. Suppose we have infinite collections $\alpha _ i, \beta _ i \in Z_ k(X)$, $i \in I$ of $k$-cycles on $X$. Suppose that the supports of $\alpha _ i$ and $\beta _ i$ form locally finite collections of closed subsets of $X$ so that $\sum \alpha _ i$ and $\sum \beta _ i$ are defined as cycles. Moreover, assume that $\alpha _ i \sim _{rat} \beta _ i$ for each $i$. Then it is not clear that $\sum \alpha _ i \sim _{rat} \sum \beta _ i$. Namely, the problem is that the rational equivalences may be given by locally finite families $\{ W_{i, j}, f_{i, j} \in R(W_{i, j})^*\} _{j \in J_ i}$ but the union $\{ W_{i, j}\} _{i \in I, j\in J_ i}$ may not be locally finite.

In many cases in practice, one has a locally finite family of closed subsets $\{ T_ i\} _{i \in I}$ of $|X|$ such that $\alpha _ i, \beta _ i$ are supported on $T_ i$ and such that $\alpha _ i \sim _{rat} \beta _ i$ “on” $T_ i$. More precisely, the families $\{ W_{i, j}, f_{i, j} \in R(W_{i, j})^*\} _{j \in J_ i}$ consist of integral closed subspaces $W_{i, j}$ with $|W_{i, j}| \subset T_ i$. In this case it is true that $\sum \alpha _ i \sim _{rat} \sum \beta _ i$ on $X$, simply because the family $\{ W_{i, j}\} _{i \in I, j\in J_ i}$ is automatically locally finite in this case.

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