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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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