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
Also, the characteristic polynomial of $\varphi '$ over $\kappa $ is the product of the characteristic polynomials of $\varphi $ and $\varphi ''$.