The Stacks project

Lemma 82.16.3. In Situation 82.2.1 let $X, Y/B$ be good. 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

\[ i_ j : W_ j \longrightarrow X \]

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

\[ p_*\left(\sum i_{j, *}\text{div}(f_ j)\right) \]

is rationally equivalent to zero on $X$.

Note that the sum is equal to

\[ \sum p_*i_{j, *}\text{div}(f_ j). \]

Let $W'_ j \subset Y$ be the integral closed subspace which is the image of $p \circ i_ j$, see Lemma 82.7.1. The collection $\{ W'_ j\} $ is locally finite in $Y$ by Lemma 82.7.5. 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 since integral closed subspace $W \subset X$ of $\delta $-dimension $k + 1$. Let $f \in R(W)^*$. Let $W' = p(W)$ as above. We get a commutative diagram of morphisms

\[ \xymatrix{ W \ar[r]_ i \ar[d]_{p'} & X \ar[d]^ p \\ W' \ar[r]^{i'} & Y } \]

Note that $p_*i_*\text{div}(f) = i'_*(p')_*\text{div}(f)$ by Lemma 82.8.2. 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. Since $(p')_*\text{div}(f)$ is a $k$-cycle, we see that $(p')_*\text{div}(f) = n[W']$ for some $n \in \mathbf{Z}$. In order to prove that $n = 0$ we may replace $W'$ by a nonempty open subspace. In particular, we may and do assume that $W'$ is a scheme. Let $\eta \in W'$ be the generic point. Let $K = \kappa (\eta ) = R(W')$ be the function field. Consider the base change diagram

\[ \xymatrix{ W_\eta \ar[r] \ar[d]_ c & W \ar[d]^{p'} \\ \mathop{\mathrm{Spec}}(K) \ar[r]^\eta & W' } \]

Observe that $c$ is proper. Also $|W_\eta |$ has dimension $1$: use Decent Spaces, Lemma 68.18.6 to identify $|W_\eta |$ as the subspace of $|W|$ of points mapping to $\eta $ and note that since $\dim _\delta (W) = k + 1$ and $\delta (\eta ) = k$ points of $W_\eta $ must have $\delta $-value either $k$ or $k + 1$. Thus the local rings have dimension $\leq 1$ by Decent Spaces, Lemma 68.20.2. By Spaces over Fields, Lemma 72.9.3 we find that $W_\eta $ is a scheme. Since $\mathop{\mathrm{Spec}}(K)$ is the limit of the nonempty affine open subschemes of $W'$ we conclude that we may assume that $W$ is a scheme by Limits of Spaces, Lemma 70.5.11. Then finally by the case of schemes (Chow Homology, Lemma 42.20.3) we find that $n = 0$.

The case $\dim _\delta (W') = k + 1$. In this case Lemma 82.14.1 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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