15.120 Determinants of endomorphisms of finite length modules
Let $(R, \mathfrak m, \kappa )$ be a local ring. Consider the category of pairs $(M, \varphi )$ consisting of a finite length $R$-module and an endomorphism $\varphi : M \to M$. This category is abelian and every object is Artinian as well as Noetherian. See Homology, Section 12.9 for definitions.
If $(M, \varphi )$ is a simple object of this category, then $M$ is annihilated by $\mathfrak m$ since otherwise $(\mathfrak m M, \varphi |_{\mathfrak m M})$ would be a nontrivial suboject. Also $\dim _\kappa (M) = \text{length}_ R(M)$ is finite. Thus we may define the determinant and the trace
\[ \det \nolimits _\kappa (\varphi ),\quad \text{Trace}_\kappa (\varphi ) \]
as elements of $\kappa $ using linear algebra. Simlarly for the characteristic polynomial of $\varphi $ in this case.
By Homology, Lemma 12.9.6 for an arbitrary object $(M, \varphi )$ of our category we have a finite filtration
\[ 0 \subset M_1 \subset \ldots \subset M_ n = M \]
by submodules stable under $\varphi $ such that $(M_ i/M_{i - 1}, \varphi _ i)$ is a simple object of the category where $\varphi _ i : M_ i/M_{i - 1} \to M_ i/M_{i - 1}$ is the induced map. We define the determinant of $(M, \varphi )$ over $\kappa $ as
\[ \det \nolimits _\kappa (\varphi ) = \prod \det \nolimits _\kappa (\varphi _ i) \]
with $\det _\kappa (\varphi _ i)$ as defined in the previous paragraph. We define the trace of $(M, \varphi )$ over $\kappa $ as
\[ \text{Trace}_\kappa (\varphi ) = \sum \text{Trace}_\kappa (\varphi _ i) \]
with $\text{Trace}_\kappa (\varphi _ i)$ as defined in the previous paragraph. We can similarly define the characteristic polynomial of $\varphi $ over $\kappa $ as the product of the characteristic polynomials of $\varphi _ i$ as defined in the previous paragraph. By Jordan-Hölder (Homology, Lemma 12.9.7) this is well defined.
Lemma 15.120.1. Let $(R, \mathfrak m, \kappa )$ be a local ring. Let $0 \to (M, \varphi ) \to (M', \varphi ') \to (M'', \varphi '') \to 0$ be a short exact sequence in the category discussed above. Then
\[ \det \nolimits _\kappa (\varphi ') = \det \nolimits _\kappa (\varphi )\det \nolimits _\kappa (\varphi ''),\quad \text{Trace}_\kappa (\varphi ') = \text{Trace}_\kappa (\varphi ) + \text{Trace}_\kappa (\varphi '') \]
Also, the characteristic polynomial of $\varphi '$ over $\kappa $ is the product of the characteristic polynomials of $\varphi $ and $\varphi ''$.
Proof.
Left as an exercise.
$\square$
Lemma 15.120.2. Let $(R, \mathfrak m, \kappa ) \to (R', \mathfrak m', \kappa ')$ be a local homomorphism of local rings. Assume that $\kappa '/\kappa $ is a finite extension. Let $u \in R'$. Then for any finite length $R'$-module $M'$ we have
\[ \det \nolimits _\kappa (u : M' \to M') = \text{Norm}_{\kappa '/\kappa }(u \bmod \mathfrak m')^ m \]
where $m = \text{length}_{R'}(M')$.
Proof.
Observe that the statement makes sense as $\text{length}_ R(M') = \text{length}_{R'}(M') [\kappa ' : \kappa ]$. If $M' = \kappa '$, then the equality holds by definition of the norm as the determinant of the linear operator given by multiplication by $u$. In general one reduces to this case by chosing a suitable filtration and using the multiplicativity of Lemma 15.120.1. Some details omitted.
$\square$
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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