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Über Differentialoperatoren und \({\mathcal D}\)-Moduln in positiver Charakteristik. (On differential operators and \({\mathcal D}\)- modules in positive characteristic). (German) Zbl 0607.14010
This paper is about differential operators and \({\mathcal D}\)-modules on a smooth variety over a field of prime characteristic. Beilinson and Bernstein called a (smooth) variety \({\mathcal D}\)-affine, if every \({\mathcal D}\)-module is generated by global sections and all its higher cohomology groups vanish. They showed, that all flag manifolds over a field of characteristic \(0\) are \({\mathcal D}\)-affine.
A smooth variety is called \({\mathcal D}\)-quasi-affine, if every \({\mathcal D}\)- module is generated by global sections. Beside some generalities the main results of the article under consideration are the \({\mathcal D}\)-affinity of the projective space, the \({\mathcal D}\)-quasi-affinity of the ordinary flag manifolds (G/B) and the \({\mathcal D}\)-affinity of the ordinary flag manifold of \(Sl_ 3\) (over a field of prime characteristic). In contrast to characteristic \(0\) generally there exists some non-vanishing higher cohomology group of the associated graded algebra gr(\({\mathcal D})\) on an ordinary flag manifold.

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14M15 Grassmannians, Schubert varieties, flag manifolds
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