Lemma 42.51.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $b : W \longrightarrow \mathbf{P}^1_ X$ be a proper morphism of schemes. Let $n \geq 1$. For $i = 1, \ldots , n$ let $Z_ i \subset X$ be a closed subscheme, let $Q_ i \in D(\mathcal{O}_ W)$ be a perfect object, let $p_ i \geq 0$ be an integer, and let $T_ i \subset W_\infty$, $i = 1, \ldots , n$ be closed. Denote $W_ i = b^{-1}(\mathbf{P}^1_{Z_ i})$. Assume

1. for $i = 1, \ldots , n$ the assumption of Lemma 42.49.1 hold for $b, Z_ i, Q_ i, T_ i, p_ i$,

2. $Q_ i|_{W \setminus W_ i}$ is zero, resp. isomorphic to a finite locally free module of rank $< p_ i$ placed in cohomological degree $0$,

3. $Q_ i$ on $W$ satisfies assumption (3) of Situation 42.50.1.

Then $P'_{p_ n}(Q_ n) \circ \ldots \circ P'_{p_1}(Q_1)$ is equal to

$(W_{n, \infty } \cap \ldots \cap W_{1, \infty } \to Z_ n \cap \ldots \cap Z_1)_* \circ P'_{p_ n}(Q_ n|_{W_{n, \infty }}) \circ \ldots \circ P'_{p_1}(Q_1|_{W_{1, \infty }}) \circ C$

in $A^{p_ n + \ldots + p_1}(Z_ n \cap \ldots \cap Z_1 \to X)$, resp. $c'_{p_ n}(Q_ n) \circ \ldots \circ c'_{p_1}(Q_1)$ is equal to

$(W_{n, \infty } \cap \ldots \cap W_{1, \infty } \to Z_ n \cap \ldots \cap Z_1)_* \circ c'_{p_ n}(Q_ n|_{W_{n, \infty }}) \circ \ldots \circ c'_{p_1}(Q_1|_{W_{1, \infty }}) \circ C$

in $A^{p_ n + \ldots + p_1}(Z_ n \cap \ldots \cap Z_1 \to X)$.

Proof. Let us prove the statement on Chern classes by induction on $n$; the statement on $P_ p(-)$ is proved in the exact same manner. The case $n = 1$ is the construction of $c'_{p_1}(Q_1)$ because $W_{1, \infty }$ is the inverse image of $Z_1$ in $W_\infty$. For $n > 1$ we have by induction that $c'_{p_ n}(Q_ n) \circ \ldots \circ c'_{p_1}(Q_1)$ is equal to

$c'_{p_ n}(Q_ n) \circ (W_{n - 1, \infty } \cap \ldots \cap W_{1, \infty } \to Z_{n - 1} \cap \ldots \cap Z_1)_* \circ c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \infty }) \circ \ldots \circ c'_{p_1}(Q_1|_{W_{1, \infty }}) \circ C$

By Lemma 42.49.2 the restriction of $c'_{p_ n}(Q_ n)$ to $Z_{n - 1} \cap \ldots \cap Z_1$ is computed by the closed subset $Z_ n \cap \ldots \cap Z_1$, the morphism $b' : W_{n - 1} \cap \ldots \cap W_1 \to \mathbf{P}^1_{Z_{n - 1} \cap \ldots \cap Z_1}$ and the restriction of $Q_ n$ to $W_{n - 1} \cap \ldots \cap W_1$. Observe that $(b')^{-1}(Z_ n) = W_ n \cap \ldots \cap W_1$ and that $(W_ n \cap \ldots \cap W_1)_\infty = W_{n, \infty } \cap \ldots \cap W_{1, \infty }$. Denote $C_{n - 1} \in A^0(W_{n - 1, \infty } \cap \ldots \cap W_{1, \infty } \to Z_{n - 1} \cap \ldots \cap Z_1)$ the class of Lemma 42.48.1. We conclude the restriction of $c'_{p_ n}(Q_ n)$ to $Z_{n - 1} \cap \ldots \cap Z_1$ is

\begin{align*} & (W_{n, \infty } \cap \ldots \cap W_{1, \infty } \to Z_ n \cap \ldots \cap Z_1)_* \circ c'_{p_ n}(Q_ n|_{(W_ n \cap \ldots \cap W_1)_\infty }) \circ C_{n - 1} \\ & = (W_{n, \infty } \cap \ldots \cap W_{1, \infty } \to Z_ n \cap \ldots \cap Z_1)_* \circ c'_{p_ n}(Q_ n|_{W_{n, \infty }}) \circ C_{n - 1} \end{align*}

where the equality follows from Lemma 42.47.3 (we omit writing the restriction on the right). Hence the above becomes

\begin{align*} (W_{n, \infty } \cap \ldots \cap W_{1, \infty } \to Z_ n \cap \ldots \cap Z_1)_* \circ c'_{p_ n}(Q_ n|_{W_ n, \infty }) \circ \\ C_{n - 1} \circ (W_{n - 1, \infty } \cap \ldots \cap W_{1, \infty } \to Z_{n - 1} \cap \ldots \cap Z_1)_* \\ \circ c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \infty }) \circ \ldots \circ c'_{p_1}(Q_1|_{W_{1, \infty }}) \circ C \end{align*}

By Lemma 42.48.4 we know that the composition $C_{n - 1} \circ (W_{n - 1, \infty } \cap \ldots \cap W_{1, \infty } \to Z_{n - 1} \cap \ldots \cap Z_1)_*$ is the identity on elements in the image of the gysin map

$(W_{n - 1, \infty } \cap \ldots \cap W_{1, \infty } \to W_{n - 1} \cap \ldots \cap W_1)^*$

Thus it suffices to show that any element in the image of $c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \infty }) \circ \ldots \circ c'_{p_1}(Q_1|_{W_{1, \infty }}) \circ C$ is in the image of the gysin map. We may write

$c'_{p_ i}(Q_ i|_{W_{i, \infty }}) = \text{restriction of } c_{p_ i}(W_ i \to W, Q_ i) \text{ to } W_{i, \infty }$

by Lemma 42.50.9 and assumptions (2) and (3) on $Q_ i$ in the statement of the lemma. Thus, if $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W)$ restricts to the flat pullback of $\alpha$ on $b^{-1}(\mathbf{A}^1_ X)$, then

\begin{align*} & c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \infty }) \cap \ldots \cap c'_{p_1}(Q_1|_{W_{1, \infty }}) \cap C \cap \alpha \\ & = c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \infty }) \cap \ldots \cap c'_{p_1}(Q_1|_{W_{1, \infty }}) \cap i_\infty ^* \beta \\ & = c_{p_{n - 1}}(W_{n - 1} \to W, Q_{n - 1}) \cap \ldots \cap c_{p_{n - 1}}(W_1 \to W, Q_1) \cap i_\infty ^* \beta \\ & = (W_{n - 1, \infty } \cap \ldots \cap W_{1, \infty } \to W_{n - 1} \cap \ldots \cap W_1)^* \left(c_{p_{n - 1}}(W_{n - 1} \to W, Q_{n - 1}) \cap \ldots \cap c_{p_1}(W_1 \to W, Q_1) \cap \beta \right) \end{align*}

as desired. Namely, for the last equality we use that $c_{p_ i}(W_ i \to W, Q_ i)$ is a bivariant class and hence commutes with $i_\infty ^*$ by definition. $\square$

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