Lemma 42.49.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme. Let

\[ b : W \longrightarrow \mathbf{P}^1_ X \]

be a proper morphism of schemes. Let $Q \in D(\mathcal{O}_ W)$ be a perfect object. Denote $W_\infty \subset W$ the inverse image of the divisor $D_\infty \subset \mathbf{P}^1_ X$ with complement $\mathbf{A}^1_ X$. We assume

Chern classes of $Q$ are defined (Section 42.46),

$b$ is an isomorphism over $\mathbf{A}^1_ X$,

there exists a closed subscheme $T \subset W_\infty $ containing all points of $W_\infty $ lying over $X \setminus Z$ such that $Q|_ T$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_ T$-module of rank $< p$ sitting in cohomological degree $0$.

Then there exists a canonical bivariant class

\[ P'_ p(Q),\text{ resp. }c'_ p(Q) \in A^ p(Z \to X) \]

with $(Z \to X)_* \circ P'_ p(Q) = P_ p(Q|_{X \times \{ 0\} })$, resp. $(Z \to X)_* \circ c'_ p(Q) = c_ p(Q|_{X \times \{ 0\} })$.

**Proof.**
Denote $E \subset W_\infty $ the inverse image of $Z$. Then $W_\infty = T \cup E$ and $b$ induces a proper morphism $E \to Z$. Denote $C \in A^0(W_\infty \to X)$ the bivariant class constructed in Lemma 42.48.1. Denote $P'_ p(Q|_ E)$, resp. $c'_ p(Q|_ E)$ in $A^ p(E \to W_\infty )$ the bivariant class constructed in Lemma 42.47.1. This makes sense because $(Q|_ E)|_{T \cap E}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{E \cap T}$-module of rank $< p$ sitting in cohomological degree $0$ by assumption (A2). Then we define

\[ P'_ p(Q) = (E \to Z)_* \circ P'_ p(Q|_ E) \circ C,\text{ resp. } c'_ p(Q) = (E \to Z)_* \circ c'_ p(Q|_ E) \circ C \]

This is a bivariant class, see Lemma 42.33.4. Since $E \to Z \to X$ is equal to $E \to W_\infty \to W \to X$ we see that

\begin{align*} (Z \to X)_* \circ c'_ p(Q) & = (W \to X)_* \circ i_{\infty , *} \circ (E \to W_\infty )_* \circ c'_ p(Q|_ E) \circ C \\ & = (W \to X)_* \circ i_{\infty , *} \circ c_ p(Q|_{W_\infty }) \circ C \\ & = (W \to X)_* \circ c_ p(Q) \circ i_{\infty , *} \circ C \\ & = (W \to X)_*\circ c_ p(Q) \circ i_{0, *} \\ & = (W \to X)_* \circ i_{0, *} \circ c_ p(Q|_{X \times \{ 0\} }) \\ & = c_ p(Q|_{X \times \{ 0\} }) \end{align*}

The second equality holds by Lemma 42.47.4. The third equality because $c_ p(Q)$ is a bivariant class. The fourth equality by Lemma 42.48.1. The fifth equality because $c_ p(Q)$ is a bivariant class. The final equality because $(W_0 \to W) \circ (W \to X)$ is the identity on $X$ if we identify $W_0$ with $X$ as we've done above. The exact same sequence of equations works to prove the property for $P'_ p(Q)$.
$\square$

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