Generators of $D$–modules in positive characteristic

Let $R=k[x_1, \dots ,x_d]$ or $R=k[[x_1,\dots,x_d]]$ be either a polynomial or a formal power series ring in a finite number of variables over a field $k$ of characteristic $p >0$ and let $D_{R|k}$ be the ring of $k$-linear differential operators of $R$. In this paper we prove that if $f$ is a non-zero element of $R$ then $R_f$, obtained from $R$ by inverting $f$, is generated as a $D_{R|k}$–module by $ \frac{1}{f}$. This is an amazing fact considering that the corresponding characteristic zero statement is very false. In fact we prove an analog of this result for a considerably wider class of rings $R$ and a considerably wider class of $D_{R|k}$-modules.

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Published 1 January 2005