Lemma 42.68.4. Let $R$ be a local ring with maximal ideal $\mathfrak m$ and residue field $\kappa $. Let $M$ be a finite length $R$-module which is annihilated by $\mathfrak m$. Let $l = \dim _\kappa (M)$. Then the map

is an isomorphism.

Lemma 42.68.4. Let $R$ be a local ring with maximal ideal $\mathfrak m$ and residue field $\kappa $. Let $M$ be a finite length $R$-module which is annihilated by $\mathfrak m$. Let $l = \dim _\kappa (M)$. Then the map

\[ \det \nolimits _\kappa (M) \longrightarrow \wedge ^ l_\kappa (M), \quad [e_1, \ldots , e_ l] \longmapsto e_1 \wedge \ldots \wedge e_ l \]

is an isomorphism.

**Proof.**
It is clear that the rule described in the lemma gives a $\kappa $-linear map since all of the admissible relations are satisfied by the usual symbols $e_1 \wedge \ldots \wedge e_ l$. It is also clearly a surjective map. Since by Lemma 42.68.3 the left hand side has dimension at most one we see that the map is an isomorphism.
$\square$

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