The Stacks project

43.8 Rational Equivalence

We are going to define rational equivalence in a way which at first glance may seem different from what you are used to, or from what is in [Chapter I, F] or Chow Homology, Section 42.19. However, in Section 43.9 we will show that the two notions agree.

Let $X$ be a variety. Let $W \subset X \times \mathbf{P}^1$ be a closed subvariety of dimension $k + 1$. Let $a, b$ be distinct closed points of $\mathbf{P}^1$. Assume that $X \times a$, $X \times b$ and $W$ intersect properly:

\[ \dim (W \cap X \times a) \leq k,\quad \dim (W \cap X \times b) \leq k. \]

This is true as soon as $W \to \mathbf{P}^1$ is dominant or if $W$ is contained in a fibre of the projection over a closed point different from $a$ or $b$ (this is an uninteresting case which we will discard). In this situation the scheme theoretic fibre $W_ a$ of the morphism $W \to \mathbf{P}^1$ is equal to the scheme theoretic intersection $W \cap X \times a$ in $X \times \mathbf{P}^1$. Identifying $X \times a$ and $X \times b$ with $X$ we may think of the fibres $W_ a$ and $W_ b$ as closed subschemes of $X$ of dimension $\leq k$1. A basic example of a rational equivalence is

\[ [W_ a]_ k \sim _{rat} [W_ b]_ k \]

The cycles $[W_ a]_ k$ and $[W_ b]_ k$ are easy to compute in practice (given $W$) because they are obtained by proper intersection with a Cartier divisor (we will see this in Section 43.17). Since the automorphism group of $\mathbf{P}^1$ is $2$-transitive we may move the pair of closed points $a, b$ to any pair we like. A traditional choice is to choose $a = 0$ and $b = \infty $.

More generally, let $\alpha = \sum n_ i [W_ i]$ be a $(k + 1)$-cycle on $X \times \mathbf{P}^1$. Let $a_ i, b_ i$ be pairs of distinct closed points of $\mathbf{P}^1$. Assume that $X \times a_ i$, $X \times b_ i$ and $W_ i$ intersect properly, in other words, each $W_ i, a_ i, b_ i$ satisfies the condition discussed above. A cycle rationally equivalent to zero is any cycle of the form

\[ \sum n_ i([W_{i, a_ i}]_ k - [W_{i, b_ i}]_ k). \]

This is indeed a $k$-cycle. The collection of $k$-cycles rationally equivalent to zero is an additive subgroup of the group of $k$-cycles. We say two $k$-cycles are rationally equivalent, notation $\alpha \sim _{rat} \alpha '$, if $\alpha - \alpha '$ is a cycle 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 $X$. We will see in Lemma 43.9.1 that this agrees with the Chow group as defined in Chow Homology, Definition 42.19.1.

[1] We will sometimes think of $W_ a$ as a closed subscheme of $X \times \mathbf{P}^1$ and sometimes as a closed subscheme of $X$. It should always be clear from context which point of view is taken.

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