The Stacks project

Lemma 41.42.7. Let $(S, \delta )$ be as in Situation 41.7.1. Let $X$ be locally of finite type over $S$. Let $i_ j : X_ j \to X$, $j = 1, 2$ be closed immersions such that $X = X_1 \cup X_2$ set theoretically. Let $E, F \in D(\mathcal{O}_ X)$ be perfect objects. Assume

  1. chern classes of $E$ and $F$ are defined,

  2. the restrictions $E|_{X_1 \cap X_2}$ and $F|_{X_1 \cap X_2}$ are isomorphic to a finite locally free $\mathcal{O}_{X_1}$-modules of rank $< p$ and $< q$ sitting in cohomological degree $0$.

With notation as in Remark 41.31.13 set

\[ c^{(p)}(E) = 1 + c_1(E) + \ldots + c_{p - 1}(E) + c'_ p(E|_{X_2}) + c'_{p + 1}(E|_{X_2}) + \ldots \in A^{(p)}(X_2 \to X) \]

with $c'_ p(E|_{X_2})$ as in Lemma 41.42.1. Similarly for $c^{(q)}(F)$ and $c^{(p + q)}(E \oplus F)$. Then $c^{(p + q)}(E \oplus F) = c^{(p)}(E)c^{(q)}(F)$ in $A^{(p + q)}(X_2 \to X)$.

Proof. Immediate from the characterization of the classes in Lemma 41.42.1 and the additivity in Lemma 41.41.4. $\square$

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