Weyl algebra in positive characteristic ($k[x_1,\dots ,x_n]/(x_1^p,\dots ,x_n^p)$)

In characteristic $p>0$, multiplication by $x_i$ and formal differentiation by $x_i$ act on the displayed vector space, because differentiation preserves the ideal generated by the $p$th powers. They satisfy the Weyl algebra relations and yield a nonzero module of dimension $p^n$. The central elements $x_i^p$ and ${\partial }_i^p$ also show why the characteristic-zero simplicity argument cannot apply. In particular the characteristic-zero Bernstein inequality for Weyl algebra modules fails for this finite-dimensional module.

// Destination: module-theory.bigb