The Stacks project

Lemma 42.32.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Assume that for every $y \in Y$, there exists an open neighbourhood $U \subset Y$ such that $f|_{f^{-1}(U)} : f^{-1}(U) \to U$ is identified with the morphism $U \times \mathbf{A}^ r \to U$. Then $f^* : \mathop{\mathrm{CH}}\nolimits _ k(Y) \to \mathop{\mathrm{CH}}\nolimits _{k + r}(X)$ is surjective for all $k \in \mathbf{Z}$.

Proof. Let $\alpha \in \mathop{\mathrm{CH}}\nolimits _{k + r}(X)$. Write $\alpha = \sum m_ j[W_ j]$ with $m_ j \not= 0$ and $W_ j$ pairwise distinct integral closed subschemes of $\delta $-dimension $k + r$. Then the family $\{ W_ j\} $ is locally finite in $X$. For any quasi-compact open $V \subset Y$ we see that $f^{-1}(V) \cap W_ j$ is nonempty only for finitely many $j$. Hence the collection $Z_ j = \overline{f(W_ j)}$ of closures of images is a locally finite collection of integral closed subschemes of $Y$.

Consider the fibre product diagrams

\[ \xymatrix{ f^{-1}(Z_ j) \ar[r] \ar[d]_{f_ j} & X \ar[d]^ f \\ Z_ j \ar[r] & Y } \]

Suppose that $[W_ j] \in Z_{k + r}(f^{-1}(Z_ j))$ is rationally equivalent to $f_ j^*\beta _ j$ for some $k$-cycle $\beta _ j \in \mathop{\mathrm{CH}}\nolimits _ k(Z_ j)$. Then $\beta = \sum m_ j \beta _ j$ will be a $k$-cycle on $Y$ and $f^*\beta = \sum m_ j f_ j^*\beta _ j$ will be rationally equivalent to $\alpha $ (see Remark 42.19.6). This reduces us to the case $Y$ integral, and $\alpha = [W]$ for some integral closed subscheme of $X$ dominating $Y$. In particular we may assume that $d = \dim _\delta (Y) < \infty $.

Hence we can use induction on $d = \dim _\delta (Y)$. If $d < k$, then $\mathop{\mathrm{CH}}\nolimits _{k + r}(X) = 0$ and the lemma holds. By assumption there exists a dense open $V \subset Y$ such that $f^{-1}(V) \cong V \times \mathbf{A}^ r$ as schemes over $V$. Suppose that we can show that $\alpha |_{f^{-1}(V)} = f^*\beta $ for some $\beta \in Z_ k(V)$. By Lemma 42.14.2 we see that $\beta = \beta '|_ V$ for some $\beta ' \in Z_ k(Y)$. By the exact sequence $\mathop{\mathrm{CH}}\nolimits _ k(f^{-1}(Y \setminus V)) \to \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _ k(f^{-1}(V))$ of Lemma 42.19.3 we see that $\alpha - f^*\beta '$ comes from a cycle $\alpha ' \in \mathop{\mathrm{CH}}\nolimits _{k + r}(f^{-1}(Y \setminus V))$. Since $\dim _\delta (Y \setminus V) < d$ we win by induction on $d$.

Thus we may assume that $X = Y \times \mathbf{A}^ r$. In this case we can factor $f$ as

\[ X = Y \times \mathbf{A}^ r \to Y \times \mathbf{A}^{r - 1} \to \ldots \to Y \times \mathbf{A}^1 \to Y. \]

Hence it suffices to do the case $r = 1$. By the argument in the second paragraph of the proof we are reduced to the case $\alpha = [W]$, $Y$ integral, and $W \to Y$ dominant. Again we can do induction on $d = \dim _\delta (Y)$. If $W = Y \times \mathbf{A}^1$, then $[W] = f^*[Y]$. Lastly, $W \subset Y \times \mathbf{A}^1$ is a proper inclusion, then $W \to Y$ induces a finite field extension $R(W)/R(Y)$. Let $P(T) \in R(Y)[T]$ be the monic irreducible polynomial such that the generic fibre of $W \to Y$ is cut out by $P$ in $\mathbf{A}^1_{R(Y)}$. Let $V \subset Y$ be a nonempty open such that $P \in \Gamma (V, \mathcal{O}_ Y)[T]$, and such that $W \cap f^{-1}(V)$ is still cut out by $P$. Then we see that $\alpha |_{f^{-1}(V)} \sim _{rat} 0$ and hence $\alpha \sim _{rat} \alpha '$ for some cycle $\alpha '$ on $(Y \setminus V) \times \mathbf{A}^1$. By induction on the dimension we win. $\square$

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