= Weyl algebra in positive characteristic
{c}
{title2=$k[x_1,\ldots,x_n]/(x_1^p,\ldots,x_n^p)$}
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
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