Lemma 81.30.3. In Situation 81.2.1 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 $A^*(X)$.

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.
$\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)