Lemma 42.51.2. Assume $(S, \delta ), X, Z, b : W \to \mathbf{P}^1_ X, Q, T, p$ satisfy the assumptions of Lemma 42.49.1. Let $F \in D(\mathcal{O}_ X)$ be a perfect object such that

1. the restriction of $Q$ to $b^{-1}(\mathbf{A}^1_ X)$ is isomorphic to the pullback of $F$,

2. $F|_{X \setminus Z}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z}$-module of rank $< p$ sitting in cohomological degree $0$, and

3. $Q$ on $W$ and $F$ on $X$ satisfy assumption (3) of Situation 42.50.1.

Then the class $P'_ p(Q)$, resp. $c'_ p(Q)$ in $A^ p(Z \to X)$ constructed in Lemma 42.49.1 is equal to $P_ p(Z \to X, F)$, resp. $c_ p(Z \to X, F)$ from Definition 42.50.3.

Proof. The assumptions are preserved by base change with a morphism $X' \to X$ locally of finite type. Hence it suffices to show that $P_ p(Z \to X, F) \cap \alpha = P'_ p(Q) \cap \alpha$, resp. $c_ p(Z \to X, F) \cap \alpha = c'_ p(Q) \cap \alpha$ for any $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$. Choose $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W)$ whose restriction to $b^{-1}(\mathbf{A}^1_ X)$ is equal to the flat pullback of $\alpha$ as in the construction of $C$ in Lemma 42.48.1. Denote $W' = b^{-1}(Z)$ and denote $E = W'_\infty \subset W_\infty$ the inverse image of $Z$ by $W_\infty \to X$. The lemma follows from the following sequence of equalities (the case of $P_ p$ is similar)

\begin{align*} c'_ p(Q) \cap \alpha & = (E \to Z)_*(c'_ p(Q|_ E) \cap i_\infty ^*\beta ) \\ & = (E \to Z)_*(c_ p(E \to W_\infty , Q|_{W_\infty }) \cap i_\infty ^*\beta ) \\ & = (W'_\infty \to Z)_*(c_ p(W' \to W, Q) \cap i_\infty ^*\beta ) \\ & = (W'_\infty \to Z)_*((i'_\infty )^*(c_ p(W' \to W, Q) \cap \beta )) \\ & = (W'_\infty \to Z)_*((i'_\infty )^*(c_ p(Z' \to X, F) \cap \beta )) \\ & = (W'_0 \to Z)_*((i'_0)^*(c_ p(Z' \to X, F) \cap \beta )) \\ & = (W'_0 \to Z)_*(c_ p(Z' \to X, F) \cap i_0^*\beta )) \\ & = c_ p(Z \to X, F) \cap \alpha \end{align*}

The first equality is the construction of $c'_ p(Q)$ in Lemma 42.49.1. The second is Lemma 42.50.9. The base change of $W' \to W$ by $W_\infty \to W$ is the morphism $E = W'_\infty \to W_\infty$. Hence the third equality holds by Lemma 42.50.4. The fourth equality, in which $i'_\infty : W'_\infty \to W'$ is the inclusion morphism, follows from the fact that $c_ p(W' \to W, Q)$ is a bivariant class. For the fith equality, observe that $c_ p(W' \to W, Q)$ and $c_ p(Z' \to X, F)$ restrict to the same bivariant class in $A^ p((b')^{-1} \to b^{-1}(\mathbf{A}^1_ X))$ by assumption (1) of the lemma which says that $Q$ and $F$ restrict to the same object of $D(\mathcal{O}_{b^{-1}(\mathbf{A}^1_ X)})$; use Lemma 42.50.4. Since $(i'_\infty )^*$ annihilates cycles supported on $W'_\infty$ (see Remark 42.29.6) we conclude the fifth equality is true. The sixth equality holds because $W'_\infty$ and $W'_0$ are the pullbacks of the rationally equivalent effective Cartier divisors $D_0, D_\infty$ in $\mathbf{P}^1_ Z$ and hence $i_\infty ^*\beta$ and $i_0^*\beta$ map to the same cycle class on $W'$; namely, both represent the class $c_1(\mathcal{O}_{\mathbf{P}^1_ Z}(1)) \cap c_ p(Z \to X, F_) \cap \beta$ by Lemma 42.29.4. The seventh equality holds because $c_ p(Z \to X, F)$ is a bivariant class. By construction $W'_0 = Z$ and $i_0^*\beta = \alpha$ which explains why the final equality holds. $\square$

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