| | |
Summary: A SURPRISING FACT ABOUT D-MODULES IN
CHARACTERISTIC p > 0
JOSEP `ALVAREZ MONTANER AND GENNADY LYUBEZNIK
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.
1. Introduction
Let k be a field and let R = k[x1, . . . , xd], or R = k[[x1, . . . , xd]] be either
a ring of polynomials or formal power series in a finite number of variables
over k. Let DR be the ring of k-linear differential operators on R. For every
f R, the natural action of DR on R extends uniquely to an action on R[1
f
]
via the standard quotient rule. Hence R[1
f
|
