Lemma 42.10.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be a short exact sequence of coherent sheaves on $X$. Assume that the $\delta $-dimension of the supports of $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{H}$ is $\leq k$. Then $[\mathcal{G}]_ k = [\mathcal{F}]_ k + [\mathcal{H}]_ k$.

**Proof.**
Follows immediately from additivity of lengths, see Algebra, Lemma 10.52.3.
$\square$

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