Summary: A SURPRISING FACT ABOUT D­MODULES IN
CHARACTERISTIC p > 0
JOSEP ‘
ALVAREZ MONTANER AND GENNADY LYUBEZNIK
Abstract. Let R = k[x 1 , . . . , x d ] be the polynomial ring in d independent
variables, where k is a field of characteristic p > 0. Let DR be the ring of
k­linear di#erential 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 DR­module by 1
f
. This is an amazing fact considering
that the corresponding characteristic zero statement is very false.
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 DR be the ring of k­linear di#erential operators on R. For every
f # R, the natural action of DR on R extends uniquely to an action on R[ 1
f ]

Source: Alvarez Montaner, Josep - Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya

Collections: Mathematics