The Stacks project

Lemma 42.68.8. Let $(R', \mathfrak m') \to (R, \mathfrak m)$ be a local ring homomorphism which induces an isomorphism on residue fields $\kappa $. Then for every finite length $R$-module the restriction $M_{R'}$ is a finite length $R'$-module and there is a canonical isomorphism

\[ \det \nolimits _{R, \kappa }(M) \longrightarrow \det \nolimits _{R', \kappa }(M_{R'}) \]

This isomorphism is functorial in $M$ and compatible with the isomorphisms $\gamma _{K \to L \to M}$ of Lemma 42.68.6 defined for $\det _{R, \kappa }$ and $\det _{R', \kappa }$.

Proof. If the length of $M$ as an $R$-module is $l$, then the length of $M$ as an $R'$-module (i.e., $M_{R'}$) is $l$ as well, see Algebra, Lemma 10.52.12. Note that an admissible sequence $x_1, \ldots , x_ l$ of $M$ over $R$ is an admissible sequence of $M$ over $R'$ as $\mathfrak m'$ maps into $\mathfrak m$. The isomorphism is obtained by mapping the symbol $[x_1, \ldots , x_ l] \in \det \nolimits _{R, \kappa }(M)$ to the corresponding symbol $[x_1, \ldots , x_ l] \in \det \nolimits _{R', \kappa }(M)$. It is immediate to verify that this is functorial for isomorphisms and compatible with the isomorphisms $\gamma $ of Lemma 42.68.6. $\square$

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