Finite ring extension

Question

in reading the proof of https://stacks.math.columbia.edu/tag/04GG (10 implies 1) I came across the following. For $R$ a local ring with residue field $\kappa$. Let $f$ be a monic polynomial over $R$ and write $\overline{f}$ for the reduction to $\kappa$. Suppose that there exists $a_0 \in \kappa$ a simple root of $\overline{f}$. I would like to understand why it follows that $R[T]/(f)$ is a finitely generated $R$-module.

Answer 1

This has nothing to do with $f$ having a root in the residue field nor $R$ being local: write $f$ explicitly: let $d=\deg f$; as $f$ is monic, $$f=T^d+c_{d-1}T^{d-1}+\dots +c_1T+c_0\qquad (c_0,c_1,\dots,c_{d-1}\in R).$$

Denote $t=T+(f)$ the congruence class of $T$ in $R[T]/(f)$. We have $$t^d=-c_{d-1}t^{d-1}-\dots -c_1t-c_0,$$ and an easy induction proves that all $t^k\;(k\ge d)$ are in the $R$-module generated by $1,t,\dots,t^{d-1}$.