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GENERATORS OF D--MODULES IN POSITIVE CHARACTERISTIC
 

Summary: GENERATORS OF D--MODULES 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 DR|k 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 DR|k --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 DR|k ­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 R|k be the ring of k­linear di#erential operators on R. For
every f # R, the natural action of D R|k 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 R|k --module. It is a remarkable fact that R f has

  

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

 

Collections: Mathematics