Lemma 45.12.3. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Then

$c^ A_ i({\mathcal E} \otimes {\mathcal L}) = \sum \nolimits _{j = 0}^ i \binom {r - i + j}{j} c^ A_{i - j}({\mathcal E}) \cup c^ A_1({\mathcal L})^ j$

Proof. By the construction of $c^ A_ i$ we may assume $\mathcal{E}$ has constant rank $r$. Let $p : P \to X$ and $p' : P' \to X$ be the projective bundle associated to $\mathcal{E}$ and $\mathcal{E} \otimes \mathcal{L}$. Then there is an isomorphism $g : P \to P'$ such that $g^*\mathcal{O}_{P'}(1) = \mathcal{O}_ P(1) \otimes p^*\mathcal{L}$. See Constructions, Lemma 27.20.1. Thus

$g^*c_1^ A(\mathcal{O}_{P'}(1)) = c_1^ A(\mathcal{O}_ P(1)) + p^*c_1^ A(\mathcal{L})$

The desired equality follows formally from this and the definition of Chern classes using equation (45.12.1.1). $\square$

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