Let $k$ be an algebraically closed field.
Let $R=k[x_1,...,x_n]$ and $\mathfrak p$ be a prime ideal of $R$.
Do we have
$R_{\mathfrak p}/(\mathfrak pR_{\mathfrak p})^n$ is a finite dimensional $k$-vector space for integer $n>0$?
Edit: By comment the above conjecture is false. So I think I miss some conditions that I deemed unnecessary.
The original question is:
Let $X$ be an abelian variety over an algebraically closed field $k$ (i.e. a complete irreducible algebraic variety over $k$ with a group law $m:X\times X\rightarrow X$ such that $m$ and the inverse map are both morphisms of varieties).
Let $e$ be the identity element.
Let $\mathfrak{m}$ be the maximal ideal of the local ring $\mathscr{O}_{X,e}$.
Then show
$\mathrm{End}(\mathscr{O}_{X,e}/\mathfrak{m}^n$) is finite dimensional $k$-vector space for $n>0$
And the question comes from page 41 in the book "Abelian Varieties" by Mumford. It is a step in a proof, which used the result without clarification.
Possible ways:
(i) Try to prove $e$ is a closed point.
(ii) $X$ is everywhere nonsingular. So every stalk is a regular local ring (i.e. $\dim\mathscr{O}_{X,e}=\dim_{\mathscr{O}_{X,e}/\mathfrak{m}}\mathfrak{m}/\mathfrak{m}^2$).