Lemma 42.47.11. In Situation 42.7.1 let $X$ be locally of finite type over $S$. Let
\[ X = X_1 \cup X_2 = X'_1 \cup X'_2 \]
be two ways of writing $X$ as a set theoretic union of closed subschemes. Let $E$, $E'$ be perfect objects of $D(\mathcal{O}_ X)$ whose Chern classes are defined. Assume that $E|_{X_1}$ and $E'|_{X'_1}$ are zero1 for $i = 1, 2$. Denote
$r = P'_0(E) \in A^0(X_2 \to X)$ and $r' = P'_0(E') \in A^0(X'_2 \to X)$,
$\gamma _ p = c'_ p(E|_{X_2}) \in A^ p(X_2 \to X)$ and $\gamma '_ p = c'_ p(E'|_{X'_2}) \in A^ p(X'_2 \to X)$,
$\chi _ p = P'_ p(E|_{X_2}) \in A^ p(X_2 \to X)$ and $\chi '_ p = P'_ p(E'|_{X'_2}) \in A^ p(X'_2 \to X)$
the classes constructed in Lemma 42.47.1. Then we have
\[ c'_1((E \otimes _{\mathcal{O}_ X}^\mathbf {L} E')|_{X_2 \cap X'_2}) = r \gamma '_1 + r' \gamma _1 \]
in $A^1(X_2 \cap X'_2 \to X)$ and
\[ c'_2((E \otimes _{\mathcal{O}_ X}^\mathbf {L} E')|_{X_2 \cap X'_2}) = r \gamma '_2 + r' \gamma _2 + {r \choose 2} (\gamma '_1)^2 + (rr' - 1) \gamma '_1\gamma _1 + {r' \choose 2} \gamma _1^2 \]
in $A^2(X_2 \cap X'_2 \to X)$ and so on for higher Chern classes. Similarly, we have
\[ P'_ p((E \otimes _{\mathcal{O}_ X}^\mathbf {L} E')|_{X_2 \cap X'_2}) = \sum \nolimits _{p_1 + p_2 = p} {p \choose p_1} \chi _{p_1} \chi '_{p_2} \]
in $A^ p(X_2 \cap X'_2 \to X)$.
Proof.
First we observe that the statement makes sense. Namely, we have $X = (X_2 \cap X'_2) \cup Y$ where $Y = (X_1 \cap X'_1) \cup (X_1 \cap X'_2) \cup (X_2 \cap X'_1)$ and the object $E \otimes _{\mathcal{O}_ X}^\mathbf {L} E'$ restricts to zero on $Y$. The actual equalities follow from the characterization of our classes in Lemma 42.47.1 and the equalities of Lemma 42.46.11. We omit the details.
$\square$
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