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GENERATORS OF DMODULES IN POSITIVE CHARACTERISTIC
 

Summary: GENERATORS OF D­MODULES IN POSITIVE
CHARACTERISTIC
JOSEP ALVAREZ-MONTANER, 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 DR|k be the ring of k-
linear 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|k­module 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.
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|k be the ring of k-linear differential operators on R. For
every f R, the natural action of DR|k on R extends uniquely to an action
on the localization Rf via the standard quotient rule. Hence Rf acquires

  

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

 

Collections: Mathematics