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