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.

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