Let $V$ be a free $R$-module of finite rank (or, to start, even a finite-dimensional $K$-vector space), $N,P \in \mathrm{End}(V)$ such that $P,N$ commute, $P$ is idempotent, and $N$ is nilpotent. Consider the endomorphism $\alpha=1+(P+N)(t-1)$ of the $R[t]$-module $V[t]$.
Question. Why is the cokernel of $\alpha$ isomorphic to the image of $P$?
This is used in Rosenberg's proof of the Bass-Heller-Swan Theorem, without proof. Actually one only wants that the cokernel of $\alpha$ is finitely generated as an $R$-module. I know how to deal with the case $N=0$, but it doesn't seem to generalize. This should just be linear algebra, but I don't know how to construct a map $\mathrm{coker}(\alpha) \to \mathrm{im}(P)$ at all.