Lemma 42.52.6. In Situation 42.7.1 let $X$ be locally of finite type over $S$. Let $Z_ i \subset X$, $i = 1, 2$ be closed subschemes. Let $F_ i$, $i = 1, 2$ be perfect objects of $D(\mathcal{O}_ X)$. Assume for $i = 1, 2$ that $F_ i|_{X \setminus Z_ i}$ is zero1 and that $F_ i$ on $X$ satisfies assumption (3) of Situation 42.50.1. Denote $r_ i = P_0(Z_ i \to X, F_ i) \in A^0(Z_ i \to X)$. Then we have

$c_1(Z_1 \cap Z_2 \to X, F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2) = r_1 c_1(Z_2 \to X, F_2) + r_2 c_1(Z_1 \to X, F_1)$

in $A^1(Z_1 \cap Z_2 \to X)$ and

\begin{align*} c_2(Z_1 \cap Z_2 \to X, F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2) & = r_1 c_2(Z_2 \to X, F_2) + r_2 c_2(Z_1 \to X, F_1) + \\ & {r_1 \choose 2} c_1(Z_2 \to X, F_2)^2 + \\ & (r_1r_2 - 1) c_1(Z_2 \to X, F_2)c_1(Z_1 \to X, F_1) + \\ & {r_2 \choose 2} c_1(Z_1 \to X, F_1)^2 \end{align*}

in $A^2(Z_1 \cap Z_2 \to X)$ and so on for higher Chern classes. Similarly, we have

$ch(Z_1 \cap Z_2 \to X, F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2) = ch(Z_1 \to X, F_1) ch(Z_2 \to X, F_2)$

in $\prod _{p \geq 0} A^ p(Z_1 \cap Z_2 \to X) \otimes \mathbf{Q}$. More precisely, we have

$P_ p(Z_1 \cap Z_2 \to X, F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2) = \sum \nolimits _{p_1 + p_2 = p} {p \choose p_1} P_{p_1}(Z_1 \to X, F_1) P_{p_2}(Z_2 \to X, F_2)$

in $A^ p(Z_1 \cap Z_2 \to X)$.

Proof. Choose proper morphisms $b_ i : W_ i \to \mathbf{P}^1_ X$ and $Q_ i \in D(\mathcal{O}_{W_ i})$ as well as closed subschemes $T_ i \subset W_{i, \infty }$ as in the construction of the localized Chern classes for $F_ i$ or more generally as in Lemma 42.51.2. Choose a commutative diagram

$\xymatrix{ W \ar[d]^{g_1} \ar[rd]^ b \ar[r]_{g_2} & W_2 \ar[d]^{b_2} \\ W_1 \ar[r]^{b_1} & \mathbf{P}^1_ X }$

where all morphisms are proper and isomorphisms over $\mathbf{A}^1_ X$. For example, we can take $W$ to be the closure of the graph of the isomorphism between $b_1^{-1}(\mathbf{A}^1_ X)$ and $b_2^{-1}(\mathbf{A}^1_ X)$. By Lemma 42.51.2 we may work with $W$, $b = b_ i \circ g_ i$, $Lg_ i^*Q_ i$, and $g_ i^{-1}(T_ i)$ to construct the localized Chern classes $c_ p(Z_ i \to X, F_ i)$. Thus we reduce to the situation described in the next paragraph.

Assume we have

1. a proper morphism $b : W \to \mathbf{P}^1_ X$ which is an isomorphism over $\mathbf{A}^1_ X$,

2. $E_ i \subset W_\infty$ is the inverse image of $Z_ i$,

3. perfect objects $Q_ i \in D(\mathcal{O}_ W)$ whose Chern classes are defined, such that

1. the restriction of $Q_ i$ to $b^{-1}(\mathbf{A}^1_ X)$ is the pullback of $F_ i$, and

2. there exists a closed subscheme $T_ i \subset W_\infty$ containing all points of $W_\infty$ lying over $X \setminus Z_ i$ such that $Q_ i|_{T_ i}$ is zero.

By Lemma 42.51.2 we have

$c_ p(Z_ i \to X, F_ i) = c'_ p(Q_ i) = (E_ i \to Z_ i)_* \circ c'_ p(Q_ i|_{E_ i}) \circ C$

and

$P_ p(Z_ i \to X, F_ i) = P'_ p(Q_ i) = (E_ i \to Z_ i)_* \circ P'_ p(Q_ i|_{E_ i}) \circ C$

for $i = 1, 2$. Next, we observe that $Q = Q_1 \otimes _{\mathcal{O}_ W}^\mathbf {L} Q_2$ satisfies (3)(a) and (3)(b) for $F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2$ and $T_1 \cup T_2$. Hence we see that

$c_ p(Z_1 \cap Z_2 \to X, F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2) = (E_1 \cap E_2 \to Z_1 \cap Z_2)_* \circ c'_ p(Q|_{E_1 \cap E_2}) \circ C$

and

$P_ p(Z_1 \cap Z_2 \to X, F_1 \otimes _{\mathcal{O}_ X}^\mathbf {L} F_2) = (E_1 \cap E_2 \to Z_1 \cap Z_2)_* \circ P'_ p(Q|_{E_1 \cap E_2}) \circ C$

by the same lemma. By Lemma 42.47.11 the classes $c'_ p(Q|_{E_1 \cap E_2})$ and $P'_ p(Q|_{E_1 \cap E_2})$ can be expanded in the correct manner in terms of the classes $c'_ p(Q_ i|_{E_ i})$ and $P'_ p(Q_ i|_{E_ i})$. Then finally Lemma 42.51.1 tells us that polynomials in $c'_ p(Q_ i|_{E_ i})$ and $P'_ p(Q_ i|_{E_ i})$ agree with the corresponding polynomials in $c'_ p(Q_ i)$ and $P'_ p(Q_ i)$ as desired. $\square$

[1] Presumably there is a variant of this lemma where we only assume $F_ i|_{X \setminus Z_ i}$ is isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z_ i}$-module of rank $< p_ i$.

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