82.25 Affine bundles

This section is the analogue of Chow Homology, Section 42.32. For an affine bundle the pullback map is surjective on Chow groups.

Lemma 82.25.1. In Situation 82.2.1 let $X, Y/B$ be good. Let $f : X \to Y$ be a quasi-compact flat morphism over $B$ of relative dimension $r$. Assume that for every $y \in Y$ we have $X_ y \cong \mathbf{A}^ r_{\kappa (y)}$. 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 subspaces of $\delta$-dimension $k + r$. Then the family $\{ W_ j\}$ is locally finite in $X$. Let $Z_ j \subset Y$ be the integral closed subspace such that we obtain a dominant morphism $W_ j \to Z_ j$ as in Lemma 82.7.1. 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 $Z_ j$ of closures of images is a locally finite collection of integral closed subspaces 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 82.15.3). 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; this is the base case of the induction. Consider a nonempty open $V \subset Y$. Suppose that we can show that $\alpha |_{f^{-1}(V)} = f^*\beta$ for some $\beta \in Z_ k(V)$. By Lemma 82.10.2 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 82.15.2 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 $d$.

In particular, by replacing $Y$ by a suitable open we may assume $Y$ is a scheme with generic point $\eta$. The isomorphism $Y_\eta \cong \mathbf{A}^ r_\eta$ extends to an isomorphism over a nonempty open $V \subset Y$, see Limits of Spaces, Lemma 70.7.1. This reduces us to the case of schemes which is Chow Homology, Lemma 42.32.1. $\square$

Lemma 82.25.2. In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let

$p : L = \underline{\mathop{\mathrm{Spec}}}(\text{Sym}^*(\mathcal{L})) \longrightarrow X$

be the associated vector bundle over $X$. Then $p^* : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _{k + 1}(L)$ is an isomorphism for all $k$.

Proof. For surjectivity see Lemma 82.25.1. Let $o : X \to L$ be the zero section of $L \to X$, i.e., the morphism corresponding to the surjection $\text{Sym}^*(\mathcal{L}) \to \mathcal{O}_ X$ which maps $\mathcal{L}^{\otimes n}$ to zero for all $n > 0$. Then $p \circ o = \text{id}_ X$ and $o(X)$ is an effective Cartier divisor on $L$. Hence by Lemma 82.24.1 we see that $o^* \circ p^* = \text{id}$ and we conclude that $p^*$ is injective too. $\square$

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