42.19 Rational equivalence
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 Example 42.19.5. 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 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 $X$.
There are many other interesting (adequate) equivalence relations. Rational equivalence is the coarsest one of them all.
The fact that we obtain a presentation of $\mathop{\mathrm{CH}}\nolimits _ k(X)$ follows immediately by comparing with Definition 42.19.1.
A very simple but important lemma is the following.
Lemma 42.19.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $U \subset X$ be an open subscheme, and denote $i : Y = X \setminus U \to X$ as a reduced closed subscheme of $X$. 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 subschemes 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. Set $W_ j' \subset X$ equal to the closure of $W_ j$. 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 supported on $Y$ and the lemma follows (see Lemma 42.14.2).
$\square$
Lemma 42.19.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $X_1, X_2 \subset X$ be closed subschemes such that $X = X_1 \cup X_2$ set theoretically. For every $k \in \mathbf{Z}$ the sequence of abelian groups
\[ \xymatrix{ \mathop{\mathrm{CH}}\nolimits _ k(X_1 \cap X_2) \ar[r] & \mathop{\mathrm{CH}}\nolimits _ k(X_1) \oplus \mathop{\mathrm{CH}}\nolimits _ k(X_2) \ar[r] & \mathop{\mathrm{CH}}\nolimits _ k(X) \ar[r] & 0 } \]
is exact. Here $X_1 \cap X_2$ is the scheme theoretic intersection and the maps are the pushforward maps with one multiplied by $-1$.
Proof.
By Lemma 42.12.3 the arrow $\mathop{\mathrm{CH}}\nolimits _ k(X_1) \oplus \mathop{\mathrm{CH}}\nolimits _ k(X_2) \to \mathop{\mathrm{CH}}\nolimits _ k(X)$ is surjective. Suppose that $(\alpha _1, \alpha _2)$ maps to zero under this map. Write $\alpha _1 = \sum n_{1, i}[W_{1, i}]$ and $\alpha _2 = \sum n_{2, i}[W_{2, i}]$. Then we obtain a locally finite collection $\{ W_ j\} _{j \in J}$ of integral closed subschemes of $X$ of $\delta $-dimension $k + 1$ and $f_ j \in R(W_ j)^*$ such that
\[ \sum n_{1, i}[W_{1, i}] + \sum n_{2, i}[W_{2, i}] = \sum (i_ j)_*\text{div}(f_ j) \]
as cycles on $X$ where $i_ j : W_ j \to X$ is the inclusion morphism. Choose a disjoint union decomposition $J = J_1 \amalg J_2$ such that $W_ j \subset X_1$ if $j \in J_1$ and $W_ j \subset X_2$ if $j \in J_2$. (This is possible because the $W_ j$ are integral.) Then we can write the equation above as
\[ \sum n_{1, i}[W_{1, i}] - \sum \nolimits _{j \in J_1} (i_ j)_*\text{div}(f_ j) = - \sum n_{2, i}[W_{2, i}] + \sum \nolimits _{j \in J_2} (i_ j)_*\text{div}(f_ j) \]
Hence this expression is a cycle (!) on $X_1 \cap X_2$. In other words the element $(\alpha _1, \alpha _2)$ is in the image of the first arrow and the proof is complete.
$\square$
Example 42.19.5. Here is a “strange” example. Suppose that $S$ is the spectrum of a field $k$ with $\delta $ as in Example 42.7.2. Suppose that $X = C_1 \cup C_2 \cup \ldots $ is an infinite union of curves $C_ j \cong \mathbf{P}^1_ k$ glued together in the following way: The point $\infty \in C_ j$ is glued transversally to the point $0 \in C_{j + 1}$ for $j = 1, 2, 3, \ldots $. Take the point $0 \in C_1$. This gives a zero cycle $[0] \in Z_0(X)$. The “strangeness” in this situation is that actually $[0] \sim _{rat} 0$! Namely we can choose the rational function $f_ j \in R(C_ j)$ to be the function which has a simple zero at $0$ and a simple pole at $\infty $ and no other zeros or poles. Then we see that the sum $\sum (i_ j)_*\text{div}(f_ j)$ is exactly the $0$-cycle $[0]$. In fact it turns out that $\mathop{\mathrm{CH}}\nolimits _0(X) = 0$ in this example. If you find this too bizarre, then you can just make sure your spaces are always quasi-compact (so $X$ does not even exist for you).
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