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The Stacks project

82.30 Additivity of Chern classes

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

Lemma 82.30.1. In Situation 82.2.1 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 82.30.2. In Situation 82.2.1 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 82.30.1 and 82.29.1. \square

Lemma 82.30.3. In Situation 82.2.1 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 82.26.9, 82.30.2, and 82.28.1. \square

Lemma 82.30.4. In Situation 82.2.1 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 82.30.2 and induction. \square


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