Lemma 42.68.34 (02Q3)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.68: Appendix A: Alternative approach to key lemma
Lemma 42.68.34 (cite)

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Lemma 42.68.34. Let $A$ be a Noetherian local ring. Let $a, b \in A$ . Let

$0 \to M \to M' \to M'' \to 0$

be a short exact sequence of $A$ -modules of dimension $1$ such that $a, b$ are nonzerodivisors on all three $A$ -modules. Then

$d_{M'}(a, b) = d_ M(a, b) d_{M''}(a, b)$

in $\kappa ^*$ .

Proof. It is easy to see that this leads to a short exact sequence of exact $(2, 1)$ -periodic complexes

$0 \to (M/abM, a, b) \to (M'/abM', a, b) \to (M''/abM'', a, b) \to 0$

Hence the lemma follows from Lemma 42.68.18. $\square$

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