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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