The Stacks project

Lemma 42.54.9. With notation as above we have

\[ o^*[C_ ZX]_ n = [C_ Z Y]_{n - 1} \]

in $\mathop{\mathrm{CH}}\nolimits _{n - 1}(Y \times _{o, C_ Y X} C_ ZX)$.

Proof. Denote $W \to \mathbf{P}^1_ X$ the blowing up of $\infty (Z)$ as in Section 42.53. Similarly, denote $W' \to \mathbf{P}^1_ X$ the blowing up of $\infty (Y)$. Since $\infty (Z) \subset \infty (Y)$ we get an opposite inclusion of ideal sheaves and hence a map of the graded algebras defining these blowups. This produces a rational morphism from $W$ to $W'$ which in fact has a canonical representative

\[ W \supset U \longrightarrow W' \]

See Constructions, Lemma 27.18.1. A local calculation (omitted) shows that $U$ contains at least all points of $W$ not lying over $\infty $ and the open subscheme $C_ Z X$ of the special fibre. After shrinking $U$ we may assume $U_\infty = C_ Z X$ and $\mathbf{A}^1_ X \subset U$. Another local calculation (omitted) shows that the morphism $U_\infty \to W'_\infty $ induces the canonical morphism $C_ Z X \to C_ Y X \subset W'_\infty $ of normal cones induced by the inclusion of ideals sheaves coming from $Z \subset Y$. Denote $W'' \subset W$ the strict transform of $\mathbf{P}^1_ Y \subset \mathbf{P}^1_ X$ in $W$. Then $W''$ is the blowing up of $\mathbf{P}^1_ Y$ in $\infty (Z)$ by Divisors, Lemma 31.33.2 and hence $(W'' \cap U)_\infty = C_ ZY$.

Consider the effective Cartier divisor $i : \mathbf{P}^1_ Y \to W'$ from (8) and its associated bivariant class $i^* \in A^1(\mathbf{P}^1_ Y \to W')$ from Lemma 42.33.3. We similarly denote $(i'_\infty )^* \in A^1(W'_\infty \to W')$ the gysin map at infinity. Observe that the restriction of $i'_\infty $ (Remark 42.33.5) to $U$ is the restriction of $i_\infty ^* \in A^1(W_\infty \to W)$ to $U$. On the one hand we have

\[ (i'_\infty )^* i^* [U]_{n + 1} = i_\infty ^* i^* [U]_{n + 1} = i_\infty ^* [(W'' \cap U)_\infty ]_{n + 1} = [C_ ZY]_ n \]

because $i_\infty ^*$ kills all classes supported over $\infty $, because $i^*[U]$ and $[W'']$ agree as cycles over $\mathbf{A}^1$, and because $C_ ZY$ is the fibre of $W'' \cap U$ over $\infty $. On the other hand, we have

\[ (i'_\infty )^* i^* [U]_{n + 1} = i^* i_\infty ^*[U]_{n + 1} = i^* [U_\infty ] = o^*[C_ YX]_ n \]

because $(i'_\infty )^*$ and $i^*$ commute (Lemma 42.30.5) and because the fibre of $i : \mathbf{P}^1_ Y \to W'$ over $\infty $ factors as $o : Y \to C_ YX$ and the open immersion $C_ YX \to W'_\infty $. The lemma follows. $\square$

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