Lemma 42.68.16 (02PM)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.68: Appendix A: Alternative approach to key lemma
Lemma 42.68.16 (cite)

Lemma 42.68.16. Let $R$ be a local ring with residue field $\kappa$ . Let $(M, \varphi , \varphi )$ be a $(2, 1)$ -periodic complex over $R$ . Assume that $M$ has finite length and that $(M, \varphi , \varphi )$ is exact. Then $\text{length}_ R(M) = 2 \text{length}_ R(\mathop{\mathrm{Im}}(\varphi ))$ and

$\det \nolimits _\kappa (M, \varphi , \varphi ) = (-1)^{\text{length}_ R(\mathop{\mathrm{Im}}(\varphi ))} = (-1)^{\frac{1}{2}\text{length}_ R(M)}$

Proof. Follows directly from the sign rule in the definitions. $\square$

The Stacks project

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