## 42.32 Affine bundles

For an affine bundle the pullback map is surjective on Chow groups.

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. $\square$

Lemma 42.32.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. 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 42.32.1. 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 42.31.1 we see that $o^* \circ p^* = \text{id}$ and we conclude that $p^*$ is injective too. $\square$

Remark 42.32.3. We will see later (Lemma 42.36.3) that if $X$ is a vector bundle of rank $r$ over $Y$ then the pullback map $\mathop{\mathrm{CH}}\nolimits _ k(Y) \to \mathop{\mathrm{CH}}\nolimits _{k + r}(X)$ is an isomorphism. This is true whenever $X \to Y$ satisfies the assumptions of Lemma 42.32.1, see [Lemma 2.2, Totaro-group]. We will sketch a proof in Remark 42.32.8 using higher chow groups.

Lemma 42.32.4. In the situation of Lemma 42.32.2 denote $o : X \to L$ the zero section (see proof of the lemma). Then we have

1. $o(X)$ is the zero scheme of a regular global section of $p^*\mathcal{L}^{\otimes -1}$,

2. $o_* : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _ k(L)$ as $o$ is a closed immersion,

3. $o^* : \mathop{\mathrm{CH}}\nolimits _{k + 1}(L) \to \mathop{\mathrm{CH}}\nolimits _ k(X)$ as $o(X)$ is an effective Cartier divisor,

4. $o^* p^* : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _ k(X)$ is the identity map,

5. $o_*\alpha = - p^*(c_1(\mathcal{L}) \cap \alpha )$ for any $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$, and

6. $o^* o_* : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _{k - 1}(X)$ is equal to the map $\alpha \mapsto - c_1(\mathcal{L}) \cap \alpha$.

Proof. Since $p_*\mathcal{O}_ L = \text{Sym}^*(\mathcal{L})$ we have $p_*(p^*\mathcal{L}^{\otimes -1}) = \text{Sym}^*(\mathcal{L}) \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes -1}$ by the projection formula (Cohomology, Lemma 20.52.2) and the section mentioned in (1) is the canonical trivialization $\mathcal{O}_ X \to \mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes -1}$. We omit the proof that the vanishing locus of this section is precisely $o(X)$. This proves (1).

Parts (2), (3), and (4) we've seen in the course of the proof of Lemma 42.32.2. Of course (4) is the first formula in Lemma 42.31.1.

Part (5) follows from the second formula in Lemma 42.31.1, additivity of capping with $c_1$ (Lemma 42.25.2), and the fact that capping with $c_1$ commutes with flat pullback (Lemma 42.26.2).

Part (6) follows from Lemma 42.30.3 and the fact that $o^*p^*\mathcal{L} = \mathcal{L}$. $\square$

Lemma 42.32.5. Let $Y$ be a scheme. Let $\mathcal{L}_ i$, $i = 1, 2$ be invertible $\mathcal{O}_ Y$-modules. Let $s$ be a global section of $\mathcal{L}_1 \otimes _{\mathcal{O}_ X} \mathcal{L}_2$. Denote $i : D \to X$ the zero scheme of $s$. Then there exists a commutative diagram

$\xymatrix{ D_1 \ar[r]_{i_1} \ar[d]_{p_1} & L \ar[d]^ p & D_2 \ar[l]^{i_2} \ar[d]^{p_2} \\ D \ar[r]^ i & Y & D \ar[l]_ i }$

and sections $s_ i$ of $p^*\mathcal{L}_ i$ such that the following hold:

1. $p^*s = s_1 \otimes s_2$,

2. $p$ is of finite type and flat of relative dimension $1$,

3. $D_ i$ is the zero scheme of $s_ i$,

4. $D_ i \cong \underline{\mathop{\mathrm{Spec}}}(\text{Sym}^*(\mathcal{L}_{1 - i}^{\otimes -1})|_ D))$ over $D$ for $i = 1, 2$,

5. $p^{-1}D = D_1 \cup D_2$ (scheme theoretic union),

6. $D_1 \cap D_2$ (scheme theoretic intersection) maps isomorphically to $D$, and

7. $D_1 \cap D_2 \to D_ i$ is the zero section of the line bundle $D_ i \to D$ for $i = 1, 2$.

Moreover, the formation of this diagram and the sections $s_ i$ commutes with arbitrary base change.

Proof. Let $p : X \to Y$ be the relative spectrum of the quasi-coherent sheaf of $\mathcal{O}_ Y$-algebras

$\mathcal{A} = \left(\bigoplus \nolimits _{a_1, a_2 \geq 0} \mathcal{L}_1^{\otimes -a_1} \otimes _{\mathcal{O}_ Y} \mathcal{L}_2^{\otimes -a_2}\right)/\mathcal{J}$

where $\mathcal{J}$ is the ideal generated by local sections of the form $st - t$ for $t$ a local section of any summand $\mathcal{L}_1^{\otimes -a_1} \otimes \mathcal{L}_2^{\otimes -a_2}$ with $a_1, a_2 > 0$. The sections $s_ i$ viewed as maps $p^*\mathcal{L}_ i^{\otimes -1} \to \mathcal{O}_ X$ are defined as the adjoints of the maps $\mathcal{L}_ i^{\otimes -1} \to \mathcal{A} = p_*\mathcal{O}_ X$. For any $y \in Y$ we can choose an affine open $V \subset Y$, say $V = \mathop{\mathrm{Spec}}(B)$, containing $y$ and trivializations $z_ i : \mathcal{O}_ V \to \mathcal{L}_ i^{\otimes -1}|_ V$. Observe that $f = s(z_1z_2) \in A$ cuts out the closed subscheme $D$. Then clearly

$p^{-1}(V) = \mathop{\mathrm{Spec}}(B[z_1, z_2]/(z_1 z_2 - f))$

Since $D_ i$ is cut out by $z_ i$ everything is clear. $\square$

Lemma 42.32.6. In the situation of Lemma 42.32.5 assume $Y$ is locally of finite type over $(S, \delta )$ as in Situation 42.7.1. Then we have $i_1^*p^*\alpha = p_1^*i^*\alpha$ in $\mathop{\mathrm{CH}}\nolimits _ k(D_1)$ for all $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(Y)$.

Proof. Let $W \subset Y$ be an integral closed subscheme of $\delta$-dimension $k$. We distinguish two cases.

Assume $W \subset D$. Then $i^*[W] = c_1(\mathcal{L}_1) \cap [W] + c_1(\mathcal{L}_2) \cap [W]$ in $\mathop{\mathrm{CH}}\nolimits _{k - 1}(D)$ by our definition of gysin homomorphisms and the additivity of Lemma 42.25.2. Hence $p_1^*i^*[W] = p_1^*(c_1(\mathcal{L}_1) \cap [W]) + p_1^*(c_1(\mathcal{L}_2) \cap [W])$. On the other hand, we have $p^*[W] = [p^{-1}(W)]_{k + 1}$ by construction of flat pullback. And $p^{-1}(W) = W_1 \cup W_2$ (scheme theoretically) where $W_ i = p_ i^{-1}(W)$ is a line bundle over $W$ by the lemma (since formation of the diagram commutes with base change). Then $[p^{-1}(W)]_{k + 1} = [W_1] + [W_2]$ as $W_ i$ are integral closed subschemes of $L$ of $\delta$-dimension $k + 1$. Hence

\begin{align*} i_1^*p^*[W] & = i_1^*[p^{-1}(W)]_{k + 1} \\ & = i_1^*([W_1] + [W_2]) \\ & = c_1(p_1^*\mathcal{L}_1) \cap [W_1] + [W_1 \cap W_2]_ k \\ & = c_1(p_1^*\mathcal{L}_1) \cap p_1^*[W] + [W_1 \cap W_2]_ k \\ & = p_1^*(c_1(\mathcal{L}_1) \cap [W]) + [W_1 \cap W_2]_ k \end{align*}

by construction of gysin homomorphisms, the definition of flat pullback (for the second equality), and compatibility of $c_1 \cap -$ with flat pullback (Lemma 42.26.2). Since $W_1 \cap W_2$ is the zero section of the line bundle $W_1 \to W$ we see from Lemma 42.32.4 that $[W_1 \cap W_2]_ k = p_1^*(c_1(\mathcal{L}_2) \cap [W])$. Note that here we use the fact that $D_1$ is the line bundle which is the relative spectrum of the inverse of $\mathcal{L}_2$. Thus we get the same thing as before.

Assume $W \not\subset D$. In this case, both $i_1^*p^*[W]$ and $p_1^*i^*[W]$ are represented by the $k - 1$ cycle associated to the scheme theoretic inverse image of $W$ in $D_1$. $\square$

Lemma 42.32.7. In Situation 42.7.1 let $X$ be a scheme locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition 42.29.1. There exists a commutative diagram

$\xymatrix{ D' \ar[r]_{i'} \ar[d]_ p & X' \ar[d]^ g \\ D \ar[r]^ i & X }$

such that

1. $p$ and $g$ are of finite type and flat of relative dimension $1$,

2. $p^* : \mathop{\mathrm{CH}}\nolimits _ k(D) \to \mathop{\mathrm{CH}}\nolimits _{k + 1}(D')$ is injective for all $k$,

3. $D' \subset X'$ is the zero scheme of a global section $s' \in \Gamma (X', \mathcal{O}_{X'})$,

4. $p^*i^* = (i')^*g^*$ as maps $\mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _ k(D')$.

Moreover, these properties remain true after arbitrary base change by morphisms $Y \to X$ which are locally of finite type.

Proof. Observe that $(i')^*$ is defined because we have the triple $(\mathcal{O}_{X'}, s', i' : D' \to X')$ as in Definition 42.29.1. Thus the statement makes sense.

Set $\mathcal{L}_1 = \mathcal{O}_ X$, $\mathcal{L}_2 = \mathcal{L}$ and apply Lemma 42.32.5 with the section $s$ of $\mathcal{L} = \mathcal{L}_1 \otimes _{\mathcal{O}_ X} \mathcal{L}_2$. Take $D' = D_1$. The results now follow from the lemma, from Lemma 42.32.6 and injectivity by Lemma 42.32.2. $\square$

Remark 42.32.8. Let $(S, \delta )$ be as in Situation 42.7.1. Let $Y$ be locally of finite type over $S$. Let $r \geq 0$. Let $f : X \to Y$ be a morphism of schemes. Assume every $y \in Y$ is contained in an open $V \subset Y$ such that $f^{-1}(V) \cong V \times \mathbf{A}^ r$ as schemes over $V$. In this remark we sketch a proof of the fact that $f^* : \mathop{\mathrm{CH}}\nolimits _ k(Y) \to \mathop{\mathrm{CH}}\nolimits _ k(X)$ is an isomorphism. First, by Lemma 42.32.1 the map is surjective. Let $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(Y)$ with $f^*\alpha = 0$. We will prove that $\alpha = 0$.

Step 1. We may assume that $\dim _\delta (Y) < \infty$. (This is immediate in all cases in practice so we suggest the reader skip this step.) Namely, any rational equivalence witnessing that $f^*\alpha = 0$ on $X$, will use a locally finite collection of integral closed subschemes of dimension $k + r + 1$. Taking the union of the closures of the images of these in $Y$ we get a closed subscheme $Y' \subset Y$ of $\dim _\delta (Y') \leq k + r + 1$ such that $\alpha$ is the image of some $\alpha ' \in \mathop{\mathrm{CH}}\nolimits _ k(Y')$ and such that $(f')^*\alpha = 0$ where $f'$ is the base change of $f$ to $Y'$.

Step 2. Assume $d = \dim _\delta (Y) < \infty$. Then we can use induction on $d$. If $d < k$, then $\alpha = 0$ and we are done; this is the base case of the induction. In general, our assumption on $f$ shows we can choose a dense open $V \subset Y$ such that $U = f^{-1}(V) = \mathbf{A}^ r_ V$. Denote $Y' \subset Y$ the complement of $V$ as a reduced closed subscheme and set $X' = f^{-1}(Y')$. Consider

$\xymatrix{ \mathop{\mathrm{CH}}\nolimits ^ M_{k + r}(U, 1) \ar[r] & \mathop{\mathrm{CH}}\nolimits _{k + r}(X') \ar[r] & \mathop{\mathrm{CH}}\nolimits _{k + r}(X) \ar[r] & \mathop{\mathrm{CH}}\nolimits _{k + r}(U) \ar[r] & 0 \\ \mathop{\mathrm{CH}}\nolimits ^ M_ k(V, 1) \ar[r] \ar[u] & \mathop{\mathrm{CH}}\nolimits _ k(Y') \ar[r] \ar[u] & \mathop{\mathrm{CH}}\nolimits _ k(Y) \ar[r] \ar[u] & \mathop{\mathrm{CH}}\nolimits _ k(V) \ar[r] \ar[u] & 0 }$

Here we use the first higher Chow groups of $V$ and $U$ and the six term exact sequences constructed in Remark 42.27.3, as well as flat pullback for these higher chow groups and compatibility of flat pullback with these six term exact sequences. Since $U = \mathbf{A}^ r_ V$ the vertical map on the right is an isomorphism. The map $\mathop{\mathrm{CH}}\nolimits _ k(Y') \to \mathop{\mathrm{CH}}\nolimits _{k + r}(X')$ is bijective by induction on $d$. Hence to finish the argument is suffices to show that

$\mathop{\mathrm{CH}}\nolimits ^ M_ k(V, 1) \longrightarrow \mathop{\mathrm{CH}}\nolimits ^ M_{k + r}(U, 1)$

is surjective. Arguing as in the proof of Lemma 42.32.1 this reduces to Step 3 below.

Step 3. Let $F$ be a field. Then $\mathop{\mathrm{CH}}\nolimits ^ M_0(\mathbf{A}^1_ F, 1) = 0$. (In the proof of the lemma cited above we proved analogously that $\mathop{\mathrm{CH}}\nolimits _0(\mathbf{A}^1_ F) = 0$.) We have

$\mathop{\mathrm{CH}}\nolimits ^ M_0(\mathbf{A}^1_ F, 1) = \mathop{\mathrm{Coker}}\left( \partial : K^ M_2(F(T)) \longrightarrow \bigoplus \nolimits _{\mathfrak p \subset F[T]\text{ maximal}} \kappa (\mathfrak p)^*\right)$

The classical argument for the vanishing of the cokernel is to show by induction on the degree of $\kappa (\mathfrak p)/F$ that the summand corresponding to $\mathfrak p$ is in the image. If $\mathfrak p$ is generated by the irreducible monic polynomial $P(T) \in F[T]$ and if $u \in \kappa (x)^*$ is the residue class of some $Q(T) \in F[T]$ with $\deg (Q) < \deg (P)$ then one shows that $\partial (Q, P)$ produces the element $u$ at $\mathfrak p$ and perhaps some other units at primes dividing $Q$ which have lower degree. This finishes the sketch of the proof.

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