The Stacks project

Lemma 42.45.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $ 0 \to \mathcal{E}_1 \to \mathcal{E} \to \mathcal{E}_2 \to 0 $ be a short exact sequence of finite locally free $\mathcal{O}_ X$-modules. Then we have the equality

\[ ch(\mathcal{E}) = ch(\mathcal{E}_1) + ch(\mathcal{E}_2) \]

More precisely, we have $P_ p(\mathcal{E}) = P_ p(\mathcal{E}_1) + P_ p(\mathcal{E}_2)$ in $A^ p(X)$ where $P_ p$ is as in Example 42.43.6.

Proof. It suffices to prove the more precise statement. By Section 42.43 this follows because if $x_{1, i}$, $i = 1, \ldots , r_1$ and $x_{2, i}$, $i = 1, \ldots , r_2$ are the Chern roots of $\mathcal{E}_1$ and $\mathcal{E}_2$, then $x_{1, 1}, \ldots , x_{1, r_1}, x_{2, 1}, \ldots , x_{2, r_2}$ are the Chern roots of $\mathcal{E}$. Hence we get the result from our choice of $P_ p$ in Example 42.43.6. $\square$

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