## 81.30 Additivity of Chern classes

This section is the analogue of Chow Homology, Section 42.40.

Lemma 81.30.1. In Situation 81.2.1 let $X/B$ be good. Let $\mathcal{E}$, $\mathcal{F}$ be finite locally free sheaves on $X$ of ranks $r$, $r - 1$ which fit into a short exact sequence

$0 \to \mathcal{O}_ X \to \mathcal{E} \to \mathcal{F} \to 0$

Then we have

$c_ r(\mathcal{E}) = 0, \quad c_ j(\mathcal{E}) = c_ j(\mathcal{F}), \quad j = 0, \ldots , r - 1$

in $A^*(X)$.

Lemma 81.30.2. In Situation 81.2.1 let $X/B$ be good. Let $\mathcal{E}$, $\mathcal{F}$ be finite locally free sheaves on $X$ of ranks $r$, $r - 1$ which fit into a short exact sequence

$0 \to \mathcal{L} \to \mathcal{E} \to \mathcal{F} \to 0$

where $\mathcal{L}$ is an invertible sheaf. Then

$c(\mathcal{E}) = c(\mathcal{L}) c(\mathcal{F})$

in $A^*(X)$.

Proof. The proof is identical to the proof of Chow Homology, Lemma 42.40.2 replacing the lemmas used there by Lemmas 81.30.1 and 81.29.1. $\square$

Lemma 81.30.3. In Situation 81.2.1 let $X/B$ be good. Suppose that $\mathcal{E}$ sits in an exact sequence

$0 \to \mathcal{E}_1 \to \mathcal{E} \to \mathcal{E}_2 \to 0$

of finite locally free sheaves $\mathcal{E}_ i$ of rank $r_ i$. The total Chern classes satisfy

$c(\mathcal{E}) = c(\mathcal{E}_1) c(\mathcal{E}_2)$

in $A^*(X)$.

Proof. The proof is identical to the proof of Chow Homology, Lemma 42.40.3 replacing the lemmas used there by Lemmas 81.26.9, 81.30.2, and 81.28.1. $\square$

Lemma 81.30.4. In Situation 81.2.1 let $X/B$ be good. Let ${\mathcal L}_ i$, $i = 1, \ldots , r$ be invertible $\mathcal{O}_ X$-modules. Let $\mathcal{E}$ be a locally free rank $\mathcal{O}_ X$-module endowed with a filtration

$0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_ r = \mathcal{E}$

such that $\mathcal{E}_ i/\mathcal{E}_{i - 1} \cong \mathcal{L}_ i$. Set $c_1({\mathcal L}_ i) = x_ i$. Then

$c(\mathcal{E}) = \prod \nolimits _{i = 1}^ r (1 + x_ i)$

in $A^*(X)$.

Proof. Apply Lemma 81.30.2 and induction. $\square$

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