Lemma 42.55.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $r$. Then

**Proof.**
By the splitting principle we can turn this into a calculation in the polynomial ring on the Chern roots $x_1, \ldots , x_ r$ of $\mathcal{E}$. See Section 42.43. Observe that

Thus the logarithm of the left hand side of the equation in the lemma is

Please notice the minus sign in front. However, we have

Hence we see that the first nonzero term in our Chern class is in degree $r$ and equal to the predicted value. $\square$

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