In this section we develop some additional tools to allow us to work more comfortably with localized Chern classes. The following lemma is a more precise version of something we've already encountered in the proofs of Lemmas 42.49.6 and 42.49.7.
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
The following lemma gives us a tremendous amount of flexibility if we want to compute the localized Chern classes of a complex.
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
the restriction of Q to b^{-1}(\mathbf{A}^1_ X) is isomorphic to the pullback of F,
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
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 fifth 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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