Lemma 42.10.4 (02QZ)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.10: Cycle associated to a coherent sheaf
Lemma 42.10.4 (cite)

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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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