The Stacks project

Lemma 42.68.15. Let $R$ be a local ring with residue field $\kappa $. Let $(M, \varphi , \psi )$ be a $(2, 1)$-periodic complex over $R$. Assume that $M$ has finite length and that $(M, \varphi , \psi )$ is exact. Then

\[ \det \nolimits _\kappa (M, \varphi , \psi ) \det \nolimits _\kappa (M, \psi , \varphi ) = 1. \]

Proof. Omitted. $\square$

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