The Stacks project

Remark 42.68.9. Let $(R, \mathfrak m, \kappa )$ be a local ring and assume either the characteristic of $\kappa $ is zero or it is $p$ and $p R = 0$. Let $M_1, \ldots , M_ n$ be finite length $R$-modules. We will show below that there exists an ideal $I \subset \mathfrak m$ annihilating $M_ i$ for $i = 1, \ldots , n$ and a section $\sigma : \kappa \to R/I$ of the canonical surjection $R/I \to \kappa $. The restriction $M_{i, \kappa }$ of $M_ i$ via $\sigma $ is a $\kappa $-vector space of dimension $l_ i = \text{length}_ R(M_ i)$ and using Lemma 42.68.8 we see that

\[ \det \nolimits _\kappa (M_ i) = \wedge _\kappa ^{l_ i}(M_{i, \kappa }) \]

These isomorphisms are compatible with the isomorphisms $\gamma _{K \to M \to L}$ of Lemma 42.68.6 for short exact sequences of finite length $R$-modules annihilated by $I$. The conclusion is that verifying a property of $\det _\kappa $ often reduces to verifying corresponding properties of the usual determinant on the category finite dimensional vector spaces.

For $I$ we can take the annihilator (Algebra, Definition 10.40.3) of the module $M = \bigoplus M_ i$. In this case we see that $R/I \subset \text{End}_ R(M)$ hence has finite length. Thus $R/I$ is an Artinian local ring with residue field $\kappa $. Since an Artinian local ring is complete we see that $R/I$ has a coefficient ring by the Cohen structure theorem (Algebra, Theorem 10.160.8) which is a field by our assumption on $R$.

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