Lemma 15.120.1 (0GSY)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.120: Determinants of endomorphisms of finite length modules
Lemma 15.120.1 (cite)

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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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