The Stacks project

81.25 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 81.25.1. In Situation 81.2.1 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 81.7.1. 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 81.15.3). 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 81.10.2 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 81.15.2 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 69.7.1. This reduces us to the case of schemes which is Chow Homology, Lemma 42.32.1. $\square$

Lemma 81.25.2. In Situation 81.2.1 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 81.25.1. 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 81.24.1 we see that $o^* \circ p^* = \text{id}$ and we conclude that $p^*$ is injective too. $\square$

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