Generators of D-modules in positive characteristic

Abstract

Let R = k[x1, . . . , xd] or R = k[[x1, . . . , xd]] 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 DR|k be the ring of klinear differential operators of R. In this paper we prove that if f is a non-zero element of R then Rf , obtained from R by inverting f, is generated as a DR|k–module by 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 DR|k-modules.