D–module Generation in Positive Characteristic via Frobenius Descent

In this note I show that for most F–finite regular rings R of positive characteristic the localization Rf at an element f ∈ R is generated by f as a DR–module. This generalizes and gives an alternative proof of te results in [1] where the result is proven for the polynomial ring, thereby answering questions raised in [1] affirmatively. The proof given here is a surprisingly simple application of Frobenius descent, a brief but thorough discussion of which is also included. Furthermore I show how essentially the same technique yields a quite general criterion for obtaining DR–module generators of a unit R[F ]–module. 1. Rf is DR–generated by f −1 Throughout this paper R will denote a noetherian regular ring. This note was created after hearing about the result of Alvarez Montaner and Lyubeznik in [1]. For the case of R = k[x1, . . . , xn] they show the following theorem: Theorem 1.1. Let R be a regular F–finite ring of positive characteristic, which is essentially of finite type over a regular local ring. Let f ∈ R be a nonzero element. Then the DR–module Rf is generated by f −1. What is surprising about this result is that in characteristic zero it is incorrect. There, the DR–generation of Rf is governed by the BernsteinSato polynomial of f , and therefore reflects the geometry of the hypersurface defined by f = 0. Thus this is another instance where DR–modules appear coarser in positive characteristic than in characteristic zero, cf. for example [4, 11]. The proof of Theorem 1.1 given here relies on two results which are central to the study of differential operators in positive characteristic. The first one is the fact that the localization Rf has finite length as a DR–module, which is implied by the following more general result of Lyubeznik. Theorem 1.2 ([12, Theorem 5.7]). Let R be regular, F–finite and essentially of finite type over a F–finite ring. Let M be a finitely generated unit R[F ]– module. Then M has finite length as a DR–module. A unit R[F ]–module is an R–module M together with an isomorphism θM : F M −→ M . Note that F : SpecR −→ SpecR is the absolute Frobenius map, which is the identity on the underlying topological space and the pth power map on the structure sheaf. In particular on global sections this is 1What we call here a finitely generated unit R[F ]–module is called an F–finite R–module in [12]