The Stacks project

Remark 42.68.14. Here is a more down to earth description of the determinant introduced above. 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. Let us abbreviate $I_\varphi = \mathop{\mathrm{Im}}(\varphi )$, $I_\psi = \mathop{\mathrm{Im}}(\psi )$ as above. Assume that $\text{length}_ R(I_\varphi ) = a$ and $\text{length}_ R(I_\psi ) = b$, so that $a + b = \text{length}_ R(M)$ by exactness. Choose admissible sequences $x_1, \ldots , x_ a \in I_\varphi $ and $y_1, \ldots , y_ b \in I_\psi $ such that the symbol $[x_1, \ldots , x_ a]$ generates $\det _\kappa (I_\varphi )$ and the symbol $[x_1, \ldots , x_ b]$ generates $\det _\kappa (I_\psi )$. Choose $\tilde x_ i \in M$ such that $\varphi (\tilde x_ i) = x_ i$. Choose $\tilde y_ j \in M$ such that $\psi (\tilde y_ j) = y_ j$. Then $\det _\kappa (M, \varphi , \psi )$ is characterized by the equality

\[ [x_1, \ldots , x_ a, \tilde y_1, \ldots , \tilde y_ b] = (-1)^{ab} \det \nolimits _\kappa (M, \varphi , \psi ) [y_1, \ldots , y_ b, \tilde x_1, \ldots , \tilde x_ a] \]

in $\det _\kappa (M)$. This also explains the sign.

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