The Stacks project

Lemma 15.120.3. Let $(R, \mathfrak m, \kappa ) \to (R', \mathfrak m', \kappa ')$ be a flat local homomorphism of local rings such that $m = \text{length}_{R'}(R'/\mathfrak mR') < \infty $. For any $(M, \varphi )$ as above, the element $\det _\kappa (\varphi )^ m$ maps to $\det _{\kappa '}(\varphi \otimes 1 : M \otimes _ R R' \to M \otimes _ R R')$ in $\kappa '$.

Proof. The flatness of $R \to R'$ assures us that short exact sequences as in Lemma 15.120.1 base change to short exact sequences over $R'$. Hence by the multiplicativity of Lemma 15.120.1 we may assume that $(M, \varphi )$ is a simple object of our category (see introduction to this section). In the simple case $M$ is annihilated by $\mathfrak m$. Choose a filtration

\[ 0 \subset I_1 \subset I_2 \subset \ldots \subset I_{m - 1} \subset R'/\mathfrak mR' \]

whose successive quotients are isomorphic to $\kappa '$ as $R'$-modules. Then we obtain the filtration

\[ 0 \subset M \otimes _\kappa I_1 \subset M \otimes _\kappa I_2 \subset \ldots \subset M \otimes _\kappa I_{m - 1} \subset M \otimes _\kappa R'/\mathfrak mR' = M \otimes _ R R' \]

whose successive quotients are isomorphic to $M \otimes _\kappa \kappa '$. Also, these submodules are invariant under $\varphi \otimes 1$. By Lemma 15.120.1 we find

\[ \det \nolimits _{\kappa '}(\varphi \otimes 1 : M \otimes _ R R' \to M \otimes _ R R') = \det \nolimits _{\kappa '}(\varphi \otimes 1 : M \otimes _\kappa \kappa ' \to M \otimes _\kappa \kappa ')^ m = \det \nolimits _\kappa (\varphi )^ m \]

The last equality holds by the compatibility of determinants of linear maps with field extensions. This proves the lemma. $\square$

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