 
Summary: GENERATORS OF DMODULES IN POSITIVE
CHARACTERISTIC
JOSEP ALVAREZMONTANER, MANUEL BLICKLE, AND GENNADY LYUBEZNIK
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 DRk be the ring of k
linear di#erential operators of R. In this paper we prove that if f is
a nonzero element of R then Rf , obtained from R by inverting f , is
generated as a DRk 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 DRk modules.
1. Introduction
Let k be a field and let R = k[x 1 , . . . , x d ], or R = k[[x 1 , . . . , x d ]] be either
a ring of polynomials or formal power series in a finite number of variables
over k. Let D Rk be the ring of klinear di#erential operators on R. For
every f # R, the natural action of D Rk on R extends uniquely to an action
on the localization R f via the standard quotient rule. Hence R f acquires
a natural structure of D Rk module. It is a remarkable fact that R f has
