The Stacks project

Lemma 42.48.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $b : W \to \mathbf{P}^1_ X$ be a proper morphism of schemes which is an isomorphism over $\mathbf{A}^1_ X$. Denote $i_\infty : W_\infty \to W$ the inverse image of the divisor $D_\infty \subset \mathbf{P}^1_ X$ with complement $\mathbf{A}^1_ X$. Then there is a canonical bivariant class

\[ C \in A^0(W_\infty \to X) \]

with the property that $i_{\infty , *}(C \cap \alpha ) = i_{0, *}\alpha $ for $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ and similarly after any base change by $X' \to X$ locally of finite type.

Proof. Given $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ there exists a $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W)$ restricting to the flat pullback of $\alpha $ on $b^{-1}(\mathbf{A}^1_ X)$, see Lemma 42.14.2. A second choice of $\beta $ differs from $\beta $ by a cycle supported on $W_\infty $, see Lemma 42.19.3. Since the normal bundle of the effective Cartier divisor $D_\infty \subset \mathbf{P}^1_ X$ of ( is trivial, the gysin homomorphism $i_\infty ^*$ kills cycle classes supported on $W_\infty $, see Remark 42.29.6. Hence setting $C \cap \alpha = i_\infty ^*\beta $ is well defined.

Since $W_\infty $ and $W_0 = X \times \{ 0\} $ are the pullbacks of the rationally equivalent effective Cartier divisors $D_0, D_\infty $ in $\mathbf{P}^1_ X$, we see that $i_\infty ^*\beta $ and $i_0^*\beta $ map to the same cycle class on $W$; namely, both represent the class $c_1(\mathcal{O}_{\mathbf{P}^1_ X}(1)) \cap \beta $ by Lemma 42.29.4. By our choice of $\beta $ we have $i_0^*\beta = \alpha $ as cycles on $W_0 = X \times \{ 0\} $, see for example Lemma 42.31.1. Thus we see that $i_{\infty , *}(C \cap \alpha ) = i_{0, *}\alpha $ as stated in the lemma.

Observe that the assumptions on $b$ are preserved by any base change by $X' \to X$ locally of finite type. Hence we get an operation $C \cap - : \mathop{\mathrm{CH}}\nolimits _ k(X') \to \mathop{\mathrm{CH}}\nolimits _ k(W'_\infty )$ by the same construction as above. To see that this family of operations defines a bivariant class, we consider the diagram

\[ \xymatrix{ & & & \mathop{\mathrm{CH}}\nolimits _*(X) \ar[d]^{\text{flat pullback}} \\ \mathop{\mathrm{CH}}\nolimits _{* + 1}(W_\infty ) \ar[r] \ar[rd]^0 & \mathop{\mathrm{CH}}\nolimits _{* + 1}(W) \ar[d]^{i_\infty ^*} \ar[rr]^{\text{flat pullback}} & & \mathop{\mathrm{CH}}\nolimits _{* + 1}(\mathbf{A}^1_ X) \ar[r] \ar@{..>}[lld]^{C \cap -} & 0 \\ & \mathop{\mathrm{CH}}\nolimits _*(W_\infty ) } \]

for $X$ as indicated and the base change of this diagram for any $X' \to X$. We know that flat pullback and $i_\infty ^*$ are bivariant operations, see Lemmas 42.33.2 and 42.33.3. Then a formal argument (involving huge diagrams of schemes and their chow groups) shows that the dotted arrow is a bivariant operation. $\square$

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