Kashiwara, Masaki; Lauritzen, Niels Local cohomology and \(\mathcal D\)-affinity in positive characteristic. (English) Zbl 1016.14009 C. R., Math., Acad. Sci. Paris 335, No. 12, 993-996 (2002). Summary: We give an example of a \(\mathcal D\)-module on a Grassmann variety in positive characteristic with non-vanishing first cohomology group. This is a counterexample to \(\mathcal D\)-affinity and the Beilinson-Bernstein equivalence for flag manifolds in positive characteristic. Cited in 4 ReviewsCited in 11 Documents MSC: 14F17 Vanishing theorems in algebraic geometry 14G15 Finite ground fields in algebraic geometry 13D45 Local cohomology and commutative rings 14M15 Grassmannians, Schubert varieties, flag manifolds Keywords:local cohomology; \({\mathcal D}\)-module; positive characteristic; \({\mathcal D}\)-affinity; flag manifolds; non-vanishing PDFBibTeX XMLCite \textit{M. Kashiwara} and \textit{N. Lauritzen}, C. R., Math., Acad. Sci. Paris 335, No. 12, 993--996 (2002; Zbl 1016.14009) Full Text: DOI arXiv References: [1] Beilinson, A.; Bernstein, J., Localisation de \(g\)-modules, C. R. Acad. Sci. Paris, 292, 15-18 (1981) · Zbl 0476.14019 [2] R. Bezrukavnikov, I. Mirković, D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, Preprint, math.RT/0205144; R. Bezrukavnikov, I. Mirković, D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, Preprint, math.RT/0205144 [3] Peskine, C.; Szpiro, L., Dimension projective finie et cohomologie locale, Inst. Hautes Études Sci. Publ. Math., 42, 47-119 (1973) · Zbl 0268.13008 [4] Haastert, B., Über Differentialoperatoren und \(D\)-Moduln in positiver Characteristik, Manuscripta Math., 58, 385-415 (1987) · Zbl 0607.14010 [5] Huneke, C.; Lyubeznik, G., On the vanishing of local cohomology modules, Invent. Math., 102, 73-95 (1990) · Zbl 0717.13011 [6] Kempf, G., The Grothendieck-Cousin complex of an induced representation, Adv. Math., 29, 310-396 (1978) · Zbl 0393.20027 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.