Some generalities on 𝒟-modules in positive characteristic

Abstract

After the ring theoretic study of dierential 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. Bgvad followed. The purpose of this paper is to complement some basics for further study. We x an algebraically closed eldk of positive characteristic p. All the varieties considered in this paper will be smooth over k unless otherwise specied. A celebrated theorem of A. Beilinson and J. Bernstein says that if a varietyX isD-ane, then the category of D(X)-modules is equivalent to its local version the category ofDX-modules that are quasicoherent over OX .I n x 1we note that the converse also holds. Inx2 we will verify the base change theorem for the direct image functor ofD-modules as in characteristic 0, that will enable us to introduce a structure of G-equivariant D-module on local cohomology modules. IfG is an ane algebraick-group acting on a variety X, we give in x3 an innitesimal criterion for an OX-module to beG-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 DG=H-modules carries over to positive characteristic for a closed subgroup schemeH ofG. x4 contains a few applications on the flag variety. Notations .B yAlgk (resp. Schk) we will denote the category ofk-algebras (resp. k-schemes). The tensor product without a subscript is always taken overk .I f Ais ak-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 ofA-modules is denoted by ModA .I f there are twok-algebra homomorphisms from A into C, one making C into a leftA-module and the other into a rightA-module, we will callC a left (resp. right) A-ring, and denote the category of left (resp. right) A-rings by ARng (resp. RngA). For ak-varietyX the category of quasicoherentOX-modules is denoted by qcX, and the category of sheaves of abelian groups onX by AbX .I f Ais a sheaf ofk-algebras onX,AMod will denote the category of leftA-modules replacing A byA above, and dene likewise ModA, etc. In