Lemma 42.68.28 (02PY)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.68: Appendix A: Alternative approach to key lemma
Lemma 42.68.28 (cite)

Lemma 42.68.28. Let $A$ be a Noetherian local ring. Let $a, b \in A$.

If $M$ is a finite $A$-module of dimension $1$ such that $a, b$ are nonzerodivisors on $M$, then $\text{length}_ A(M/abM) < \infty $ and $(M/abM, a, b)$ is a $(2, 1)$-periodic exact complex.
If $a, b$ are nonzerodivisors and $\dim (A) = 1$ then $\text{length}_ A(A/(ab)) < \infty $ and $(A/(ab), a, b)$ is a $(2, 1)$-periodic exact complex.

In particular, in these cases $\det _\kappa (M/abM, a, b) \in \kappa ^*$, resp. $\det _\kappa (A/(ab), a, b) \in \kappa ^*$ are defined.

Proof. Follows from Lemma 42.68.27. $\square$

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