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. $\square$

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