Lemma 42.68.33 (02Q2)—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 42.68.33 (cite)

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Lemma 42.68.33. Let $A$ be a Noetherian local ring. Let $M$ be a finite $A$ -module of dimension $1$ . Let $b \in A$ be a nonzerodivisor on $M$ , and let $u \in A^*$ . Then

$d_ M(u, b) = u^{\text{length}_ A(M/bM)} \bmod \mathfrak m_ A.$

In particular, if $M = A$ , then $d_ A(u, b) = u^{\text{ord}_ A(b)} \bmod \mathfrak m_ A$ .

Proof. Note that in this case $M/ubM = M/bM$ on which multiplication by $b$ is zero. Hence $d_ M(u, b) = \det _\kappa (u|_{M/bM})$ by Lemma 42.68.17. The lemma then follows from Lemma 42.68.10. $\square$

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