Let $f:M\to N$ be a homomorphism of finite locally free $R[t]$-modules, where $R$ is a ring, and assume $P = \operatorname{coker} f$ is a f.g. $R$-module. I want to show that $P$ is finitely presented over $R$. Since $t$ is an endomorphism of the $R$-module $P$, by Cayley-Hamilton, there exists a minimal polynomial $g\in R[t]$ such that $g$ annihilates $P$. Thus, we have the following exact sequence $$M/gM\to N/gN\to P\to 0.$$
I'm wondering if $M/gM$ and $N/gN$ are both finite free $R$-modules. If $M,N$ are just finite free $R$-modules, then it's clear as $R[t]/(g)\cong R^{\oplus \deg g}$. I'm wondering how to deal with the general situation.
