Remark 42.43.5. The equalities proven above remain true even when we work with finite locally free $\mathcal{O}_ X$-modules whose rank is allowed to be nonconstant. In fact, we can work with polynomials in the rank and the Chern classes as follows. Consider the graded polynomial ring $\mathbf{Z}[r, c_1, c_2, c_3, \ldots ]$ where $r$ has degree $0$ and $c_ i$ has degree $i$. Let

$P \in \mathbf{Z}[r, c_1, c_2, c_3, \ldots ]$

be a homogeneous polynomial of degree $p$. Then for any finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ on $X$ we can consider

$P(\mathcal{E}) = P(r(\mathcal{E}), c_1(\mathcal{E}), c_2(\mathcal{E}), c_3(\mathcal{E}), \ldots ) \in A^ p(X)$

see Remark 42.38.10 for notation and conventions. To prove relations among these polynomials (for multiple finite locally free modules) we can work locally on $X$ and use the splitting principle as above. For example, we claim that

$c_2(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{E})) = P(\mathcal{E})$

where $P = 2rc_2 - (r - 1)c_1^2$. Namely, since $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{E}) = \mathcal{E} \otimes \mathcal{E}^\vee$ this follows easily from Lemmas 42.43.3 and 42.43.4 above by decomposing $X$ into parts where the rank of $\mathcal{E}$ is constant as in Remark 42.38.10.

Post a comment

Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FA7. Beware of the difference between the letter 'O' and the digit '0'.