SOME GENERALITIES ON D - MODULES IN POSITIVECHARACTERISTICMasaharu

After the ring theoretic study of diierential operators in positive characteristic by S.U. Chase and S.P. Smith, B. Haastert started investigation of D-modules on smooth varieties in positive characteristic, and the work of R. BBgvad followed. The purpose of this paper is to complement some basics for further study. We x an algebraically closed eld k of positive characteristic p. All the varieties considered in this paper will be smooth over k unless otherwise speciied. A celebrated theorem of A. Beilinson and J. Bernstein says that if a variety X is D-aane, then the category of D(X)-modules is equivalent to its local version the category of D X-modules that are quasicoherent over O X. In x1 we note that the converse also holds. In x2 we will verify the base change theorem for the direct image functor of D-modules as in characteristic 0, that will enable us to introduce a structure of G-equivariant D-module on local cohomology modules. If G is an aane algebraic k-group acting on a variety X, we give in x3 an innnitesimal criterion for an O X-module to be G-equivariant, introduce two G-equivariant versions of Haastert's X 1-modules, and show that the equivalence in characteristic 0 of the category of Harish-Chandra (Dist(G); H)-modules to the category of quasi-G-equivariant D G=H-modules carries over to positive characteristic for a closed subgroup scheme H of G. x4 contains a few applications on the ag variety. Notations. By Alg k (resp. Sch k) we will denote the category of k-algebras (resp. k-schemes). The tensor product without a subscript is always taken over k. If A is a k-algebra, AMod (resp. ModA) will denote the category of left (resp. right) A-modules. If A is commutative and if there is no need to distinguish left and right, the category of A-modules is denoted by Mod A. If there are two k-algebra homomorphisms from A into C, one making C into a left A-module and the other into a right A-module, we will call C a left (resp. right) A-ring, and denote the category of left (resp. right) A-rings by ARng (resp. RngA). For a k-variety X the category of quasicoherent O X-modules is denoted by qc X , and the category of sheaves of abelian groups on X by Ab X. If A is a sheaf of k-algebras on X, AMod will denote the category of left A-modules replacing A by A …