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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.

MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14M15 Grassmannians, Schubert varieties, flag manifolds
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References:
[1] H. H. ANDERSEN, J. C. JANTZEN: Cohomology of induced representations for algebraic groups. Math. Ann.269, 487-525 (1984) · Zbl 0529.20027 · doi:10.1007/BF01450762
[2] A. BEILINSON, J. BERNSTEIN: Localisation de g-modules. C. R. Acad. Sci.292, 15-18 (1981) · Zbl 0476.14019
[3] I. BERSTEIN: On the dimension of modules and algebras IX, Direct limits. Nagoya Math. J.13, 83-84 (1958) · Zbl 0084.26602
[4] W. BORHO, J. L. BRYLINSKI: Differential operators on homogeneous spaces III, Characteristic varieties of Harish-Chandra modules and primitive ideals. Invent. Math.80, 1-68 (1985) · Zbl 0577.22014 · doi:10.1007/BF01388547
[5] W. BORHO, H. KRAFT: Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen. Comment. Math. Helv.54, 61-104 (1979) · Zbl 0395.14013 · doi:10.1007/BF02566256
[6] N. BOURBAKI: Eléments de Mathematique, Algèbre Chap. 1-3, Paris: Herrmann 1970
[7] J. L. BRYLINSKI: Differential operators on the flag varieties. In: Young tableaux and Schur functors in algebra and geometry. Asterisque87-88, 43-60 (1981)
[8] J. L. BRYLINSKI, M. KASHIWARA: Kazhdan-Lusztig conjecture and holonomic systems. Invent. Math.64, 387-410 (1981) · Zbl 0473.22009 · doi:10.1007/BF01389272
[9] S. U. CHASE: On the homological dimension of algebras of differential operators. Commun. Algebra1 (5), 351-363 (1974) · Zbl 0315.13011 · doi:10.1080/00927877408548623
[10] E. CLINE, B. PARSHALL, L. SCOTT: A Mackey imprimitivity theory for algebraic groups. Math. Z.182, 447-471 (1983) · Zbl 0537.14029 · doi:10.1007/BF01215476
[11] R. ELKIK: Désingularisation des adhérences d’orbites polarisables et de nappes dans les algèbres de Lie réductives. Preprint, Paris (1978)
[12] W. FULTON: Intersection theory. Berlin-Heidelberg-New York: Springer Verlag 1984 · Zbl 0541.14005
[13] A. GROTHENDIECK, J. DIEUDONNE: Eléments de Géométrie Algébrique, EGA IV: Etude locale des schemas et de morphismes de schémas. Publ. Math. Inst. Hautes Etud. Sci.20 (1964),24 (1965),28 (1966),32 (1967)
[14] B. Haastert: Über Differentialoperatoren undD-Moduln in positiver Charakteristik. Dissertation, Hamburg (1986) · Zbl 0619.14011
[15] W. J. HABOUSH: Central differential operators on split semisimple groups over fields of positive characteristic. In: Séminaire d’Algèbre P. Dubreil et M.-P. Malliavin, Lecture Notes in Math.795, Berlin-Heidelberg-New York: Springer Verlag 1979 · Zbl 0437.22014
[16] R. HARTSHORNE: Algebraic geometry. Berlin-Heidelberg-New York: Springer Verlag 1977 · Zbl 0367.14001
[17] R. HARTSHORNE: Ample subvarieties of algebraic varieties. Lecture Notes in Math.156, Berlin-Heidelberg-New York, Springer Verlag 1970 · Zbl 0208.48901
[18] J. E. HUMPHREYS: Linear algebraic groups. Berlin-Heidelberg-New York: Springer Verlag 1975
[19] J. E. HUMPHREYS: Modular representations of classical Lie algebras and semisimple groups. J. Algebra19, 51-79 (1971) · Zbl 0219.17003 · doi:10.1016/0021-8693(71)90115-3
[20] J. C. JANTZEN: Über Darstellungen höherer Frobenius-Kerne halbeinfacher algebraischer Gruppen. Math. Z.164, 271-292 (1979) · Zbl 0396.20029 · doi:10.1007/BF01182273
[21] J. C. JANTZEN: Representations of algebraic groups. Erscheint 1987 bei Academic Press.
[22] S. P. SMITH: Differential operators on commutative algebras. In: Ring theory. Lecture Notes in Math.1197, Berlin-Heidelberg-New York: Springer Verlag 1986
[23] S. P. SMITH: Differential operators on the affine and projective lines in characteristic p>0. Erscheint im Séminaire d’Algèbre P. Dubreil et M.-P. Malliavin.
[24] S. P. SMITH: The global homological dimension of the ring of differential operators on a non-singular variety over a field of positive characteristic. Erscheint im J. Algebra. · Zbl 0617.13007
[25] T. A. SPRINGER: Linear algebraic groups. Boston-Basel-Stuttgart: Birkhäuser Verlag 1981
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