A short proof of the Bernstein inequality for formal power series

Abstract:

Let $k$ be a field of characteristic zero, let $R$ be the ring of formal power series in $n$ variables over $k$, and let $D(R,k)$ be the ring of $k$-linear differential operators on $R$. If $M$ is a finitely generated $D(R,k)-$module, then $d(M)\geq n$ where $d(M)$ is the dimension of $M$. This inequality is called the Bernstein inequality. We provide a short proof.