Section 43.9 (0AZH): Rational equivalence and rational functions

Let

$X$ be a variety. Let

$W \subset X$ be a subvariety of dimension

$k + 1$ . Let

$f \in \mathbf{C}(W)^*$ be a nonzero rational function on

$W$ . For every subvariety

$Z \subset W$ of dimension

$k$ one can define the order of vanishing

$\text{ord}_{W, Z}(f)$ of

$f$ at

$Z$ . If

$f$ is an element of the local ring

$\mathcal{O}_{W, Z}$ , then one has

where

$\mathcal{O}_{X, Z}$ , resp.

$\mathcal{O}_{W, Z}$ is the local ring of

$X$ , resp.

$W$ at the generic point of

$Z$ . In general one extends the definition by multiplicativity. The principal divisor associated to $f$ is

$Z_ k(W)$ . Since

$W \subset X$ is a closed subvariety we may think of

$\text{div}_ W(f)$ as a cycle on

$X$ . See Chow Homology, Section 42.17.

Lemma 43.9.1. Let $X$ be a variety. Let $W \subset X$ be a subvariety of dimension $k + 1$ . Let $f \in \mathbf{C}(W)^*$ be a nonzero rational function on $W$ . Then $\text{div}_ W(f)$ is rationally equivalent to zero on $X$ . Conversely, these principal divisors generate the abelian group of cycles rationally equivalent to zero on $X$ .

Proof. The first assertion follows from Chow Homology, Lemma 42.18.2. More precisely, let $W' \subset X \times \mathbf{P}^1$ be the closure of the graph of $f$ . Then $\text{div}_ W(f) = [W'_0]_ k - [W'_\infty ]$ in $Z_ k(W) \subset Z_ k(X)$ , see part (6) of Chow Homology, Lemma 42.18.2.

For the second, let $W' \subset X \times \mathbf{P}^1$ be a closed subvariety of dimension $k + 1$ which dominates $\mathbf{P}^1$ . We will show that $[W'_0]_ k - [W'_\infty ]_ k$ is a principal divisor which will finish the proof. Let $W \subset X$ be the image of $W'$ under the projection to $X$ . Then $W \subset X$ is a closed subvariety and $W' \to W$ is proper and dominant with fibres of dimension $0$ or $1$ . If $\dim (W) = k$ , then $W' = W \times \mathbf{P}^1$ and we see that $[W'_0]_ k - [W'_\infty ]_ k = [W] - [W] = 0$ . If $\dim (W) = k + 1$ , then $W' \to W$ is generically finite ¹. Let $f$ denote the projection $W' \to \mathbf{P}^1$ viewed as an element of $\mathbf{C}(W')^*$ . Let $g = \text{Nm}(f) \in \mathbf{C}(W)^*$ be the norm. By Chow Homology, Lemma 42.18.1 we have

$\text{div}_ W(g) = \text{pr}_{X, *}\text{div}_{W'}(f)$

Since $\text{div}_{W'}(f) = [W'_0]_ k - [W'_\infty ]_ k$ by Chow Homology, Lemma 42.18.2 the proof is complete. $\square$

[1] If

$W' \to W$ is birational, then the result follows from Chow Homology, Lemma 42.18.2. Our task is to show that even if

$W' \to W$ has degree

$> 1$ the basic rational equivalence

$[W'_0]_ k \sim _{rat} [W'_\infty ]_ k$ comes from a principal divisor on a subvariety of

$X$ .

The Stacks project

43.9 Rational equivalence and rational functions

Comments (2)

Post a comment