Definition 42.68.13. 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. The determinant of $(M, \varphi , \psi )$ is the element
\[ \det \nolimits _\kappa (M, \varphi , \psi ) \in \kappa ^* \]
such that the composition
\[ \det \nolimits _\kappa (M) \xrightarrow {\gamma _\psi \circ \sigma \circ \gamma _\varphi ^{-1}} \det \nolimits _\kappa (M) \]
is multiplication by $(-1)^{\text{length}_ R(I_\varphi )\text{length}_ R(I_\psi )} \det \nolimits _\kappa (M, \varphi , \psi )$.
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