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 $X$.
43.9 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
\[ \text{ord}_{W, Z}(f) = \text{length}_{\mathcal{O}_{X, Z}} \mathcal{O}_{W, Z}/f\mathcal{O}_{W, Z} \]
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
\[ \text{div}_ W(f) = \sum \text{ord}_{W, Z}(f)[Z] \]
in $Z_ k(W)$. 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.
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 finite1. 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. $\square$
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