The Stacks project

Lemma 81.31.3. In Situation 81.2.1 let $X/B$ be good. Let $\mathcal{E}$ and $\mathcal{F}$ be a finite locally free $\mathcal{O}_ X$-modules of ranks $r$ and $s$. Then we have

\[ c_1(\mathcal{E} \otimes \mathcal{F}) = r c_1(\mathcal{F}) + s c_1(\mathcal{E}) \]
\[ c_2(\mathcal{E} \otimes \mathcal{F}) = r^2 c_2(\mathcal{F}) + rs c_1(\mathcal{F})c_1(\mathcal{E}) + s^2 c_2(\mathcal{E}) \]

and so on (see proof).

Proof. Arguing exactly as in the proof of Lemma 81.31.2 we may assume we have invertible $\mathcal{O}_ X$-modules ${\mathcal L}_ i$, $i = 1, \ldots , r$ ${\mathcal N}_ i$, $i = 1, \ldots , s$ filtrations

\[ 0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_ r = \mathcal{E} \quad \text{and}\quad 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \mathcal{F}_2 \subset \ldots \subset \mathcal{F}_ s = \mathcal{F} \]

such that $\mathcal{E}_ i/\mathcal{E}_{i - 1} \cong \mathcal{L}_ i$ and such that $\mathcal{F}_ j/\mathcal{F}_{j - 1} \cong \mathcal{N}_ j$. Ordering pairs $(i, j)$ lexicographically we obtain a filtration

\[ 0 \subset \ldots \subset \mathcal{E}_ i \otimes \mathcal{F}_ j + \mathcal{E}_{i - 1} \otimes \mathcal{F} \subset \ldots \subset \mathcal{E} \otimes \mathcal{F} \]

with successive quotients

\[ \mathcal{L}_1 \otimes \mathcal{N}_1, \mathcal{L}_1 \otimes \mathcal{N}_2, \ldots , \mathcal{L}_1 \otimes \mathcal{N}_ s, \mathcal{L}_2 \otimes \mathcal{N}_1, \ldots , \mathcal{L}_ r \otimes \mathcal{N}_ s \]

By Lemma 81.30.4 we have

\[ c(\mathcal{E}) = \prod (1 + x_ i), \quad c(\mathcal{F}) = \prod (1 + y_ j), \quad \text{and}\quad c(\mathcal{F}) = \prod (1 + x_ i + y_ j), \]

in $A^*(X)$. The result follows from a formal computation which we omit. $\square$

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