Lemma 42.20.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be schemes locally of finite type over $S$. Let $p : X \to Y$ be a proper morphism. Suppose $\alpha , \beta \in Z_ k(X)$ are rationally equivalent. Then $p_*\alpha $ is rationally equivalent to $p_*\beta $.

**Proof.**
What do we have to show? Well, suppose we are given a collection

of closed immersions, with each $W_ j$ integral of $\delta $-dimension $k + 1$ and rational functions $f_ j \in R(W_ j)^*$. Moreover, assume that the collection $\{ i_ j(W_ j)\} _{j \in J}$ is locally finite on $X$. Then we have to show that

is rationally equivalent to zero on $X$.

Note that the sum is equal to

Let $W'_ j \subset Y$ be the integral closed subscheme which is the image of $p \circ i_ j$. The collection $\{ W'_ j\} $ is locally finite in $Y$ by Lemma 42.11.2. Hence it suffices to show, for a given $j$, that either $p_*i_{j, *}\text{div}(f_ j) = 0$ or that it is equal to $i'_{j, *}\text{div}(g_ j)$ for some $g_ j \in R(W'_ j)^*$.

The arguments above therefore reduce us to the case of a single integral closed subscheme $W \subset X$ of $\delta $-dimension $k + 1$. Let $f \in R(W)^*$. Let $W' = p(W)$ as above. We get a commutative diagram of morphisms

Note that $p_*i_*\text{div}(f) = i'_*(p')_*\text{div}(f)$ by Lemma 42.12.2. As explained above we have to show that $(p')_*\text{div}(f)$ is the divisor of a rational function on $W'$ or zero. There are three cases to distinguish.

The case $\dim _\delta (W') < k$. In this case automatically $(p')_*\text{div}(f) = 0$ and there is nothing to prove.

The case $\dim _\delta (W') = k$. Let us show that $(p')_*\text{div}(f) = 0$ in this case. Let $\eta \in W'$ be the generic point. Note that $c : W_\eta \to \mathop{\mathrm{Spec}}(K)$ is a proper integral curve over $K = \kappa (\eta )$ whose function field $K(W_\eta )$ is identified with $R(W)$. Here is a diagram

Let us denote $f_\eta \in K(W_\eta )^*$ the rational function corresponding to $f \in R(W)^*$. Moreover, the closed points $\xi $ of $W_\eta $ correspond $1 - 1$ to the closed integral subschemes $Z = Z_\xi \subset W$ of $\delta $-dimension $k$ with $p'(Z) = W'$. Note that the multiplicity of $Z_\xi $ in $\text{div}(f)$ is equal to $\text{ord}_{\mathcal{O}_{W_\eta , \xi }}(f_\eta )$ simply because the local rings $\mathcal{O}_{W_\eta , \xi }$ and $\mathcal{O}_{W, \xi }$ are identified (as subrings of their fraction fields). Hence we see that the multiplicity of $[W']$ in $(p')_*\text{div}(f)$ is equal to the multiplicity of $[\mathop{\mathrm{Spec}}(K)]$ in $c_*\text{div}(f_\eta )$. By Lemma 42.18.3 this is zero.

The case $\dim _\delta (W') = k + 1$. In this case Lemma 42.18.1 applies, and we see that indeed $p'_*\text{div}(f) = \text{div}(g)$ for some $g \in R(W')^*$ as desired. $\square$

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