A surprising fact about D-modules in characteristic p>0

Abstract

Let R = k[x1, . . . , xd] be the polynomial ring in d independent variables, where k is a field of characteristic p > 0. Let DR be the ring of k-linear differential operators of R and let f be a polynomial in R. In this work we prove that the localization R[ 1 f ] obtained from R by inverting f is generated as a DR-module by 1 f . This is an amazing fact considering that the corresponding characteristic zero statement is very false.