Lemma 42.68.7. Let $(R, \mathfrak m, \kappa )$ be any local ring. The functor

$\det \nolimits _\kappa : \left\{ \begin{matrix} \text{finite length }R\text{-modules} \\ \text{with isomorphisms} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} 1\text{-dimensional }\kappa \text{-vector spaces} \\ \text{with isomorphisms} \end{matrix} \right\}$

endowed with the maps $\gamma _{K \to L \to M}$ is characterized by the following properties

1. its restriction to the subcategory of modules annihilated by $\mathfrak m$ is isomorphic to the usual determinant functor (see Lemma 42.68.4), and

2. (1), (2) and (3) of Lemma 42.68.6 hold.

Proof. Omitted. $\square$

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