The Stacks project

Proof. Let $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W)$. Denote $i_0 : X = X \times \{ 0\} \to W$ the closed immersion of the fibre over $0$ in $\mathbf{P}^1$. Then $(W_\infty \to X)_* i_\infty ^* \beta = i_0^*\beta$ in $\mathop{\mathrm{CH}}\nolimits _ k(X)$ because $i_{\infty , *}i_\infty ^*\beta$ and $i_{0, *}i_0^*\beta$ represent the same class on $W$ (for example by Lemma 42.29.4) and hence pushforward to the same class on $X$. The restriction of $\beta$ to $b^{-1}(\mathbf{A}^1_ X)$ restricts to the flat pullback of $i_0^*\beta = (W_\infty \to X)_* i_\infty ^* \beta$ because we can check this after pullback by $i_0$, see Lemmas 42.32.2 and 42.32.4. Hence we may use $\beta$ when computing the image of $(W_\infty \to X)_*i_\infty ^*\beta$ under $C$ and we get the desired result. $\square$