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.
$\square$

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