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
Then we have
in A^*(X).
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
Then we have
in A^*(X).
Proof. The proof is identical to the proof of Chow Homology, Lemma 42.40.1 replacing the lemmas used there by Lemmas 82.26.9, 82.24.1, 82.19.4, and 82.28.1. \square
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
where \mathcal{L} is an invertible sheaf. Then
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
of finite locally free sheaves \mathcal{E}_ i of rank r_ i. The total Chern classes satisfy
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
such that \mathcal{E}_ i/\mathcal{E}_{i - 1} \cong \mathcal{L}_ i. Set c_1({\mathcal L}_ i) = x_ i. Then
in A^*(X).
Proof. Apply Lemma 82.30.2 and induction. \square
Comments (0)