The Stacks project

Lemma 81.6.4. In Situation 81.2.1 let $X/B$ be good. Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be a short exact sequence of coherent $\mathcal{O}_ X$-modules. Assume that the $\delta $-dimension of the supports of $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{H}$ are $\leq k$. Then $[\mathcal{G}]_ k = [\mathcal{F}]_ k + [\mathcal{H}]_ k$.

Proof. Let $Z$ be an integral closed subspace of $X$ with $\dim _\delta (Z) = k$. It suffices to show that the coefficients of $Z$ in $[\mathcal{G}]_ k$, $[\mathcal{F}]_ k$, and $[\mathcal{H}]_ k$ satisfy the corresponding additivity. By Lemma 81.6.2 it suffices to show

\[ \text{the length of }\mathcal{G}\text{ at }x = \text{the length of }\mathcal{F}\text{ at }x + \text{the length of }\mathcal{H}\text{ at }x \]

for any $x \in |X|$. Looking at Definition 81.4.2 this follows immediately from additivity of lengths, see Algebra, Lemma 10.52.3. $\square$


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