The Stacks project

Lemma 42.46.10. In Situation 42.7.1 let $X$ be locally of finite type over $S$. Let $E$ be a perfect object of $D(\mathcal{O}_ X)$ whose Chern classes are defined. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Then

\[ c_ i(E \otimes \mathcal{L}) = \sum \nolimits _{j = 0}^ i \binom {r - i + j}{j} c_{i - j}(E) c_1(\mathcal{L})^ j \]

provided $E$ has constant rank $r \in \mathbf{Z}$.

Proof. In the case where $E$ is locally free of rank $r$ this is Lemma 42.39.1. The reader can deduce the lemma from this special case by a formal computation. An alternative is to use the splitting principle of Remark 42.46.8. In this case one ends up having to prove the following algebra fact: if we write formally

\[ \frac{\prod _{a = 1, \ldots , n} (1 + x_ a)}{\prod _{n = 1, \ldots , m} (1 + y_ b)} = 1 + c_1 + c_2 + c_3 + \ldots \]

with $c_ i$ homogeneous of degree $i$ in $\mathbf{Z}[x_ i, y_ j]$ then we have

\[ \frac{\prod _{a = 1, \ldots , n} (1 + x_ a + t)}{\prod _{b = 1, \ldots , m} (1 + y_ b + t)} = \sum \nolimits _{i \geq 0} \sum \nolimits _{j = 0}^ i \binom {r - i + j}{j} c_{i - j} t^ j \]

where $r = n - m$. We omit the details. $\square$

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