Comptes Rendus
no. 12
Local cohomology and 𝒟-affinity in positive characteristic
[Cohomologie locale et 𝒟-affinité en caractéristique positive]
Masaki Kashiwara1 ; Niels Lauritzen2
1 Research Institute for Mathematical Sciences, Kyoto University, Japan
2 Institut for matematiske fag, Aarhus Universitet, Århus, Denmark
Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 993-996.

On donne un exemple d'un 𝒟-module sur une variété grassmannienne en caractéristique positive avec premier groupe de cohomologie non nul. On obtient ainsi un contre-exemple à la 𝒟-affinité et à équivalence de Beilinson–Bernstein pour les variétés des drapeaux en caractéristique positive.

We give an example of a 𝒟-module on a Grassmann variety in positive characteristic with non-vanishing first cohomology group. This is a counterexample to 𝒟-affinity and the Beilinson–Bernstein equivalence for flag manifolds in positive characteristic.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02616-X
Masaki Kashiwara 1 ; Niels Lauritzen 2

1 Research Institute for Mathematical Sciences, Kyoto University, Japan
2 Institut for matematiske fag, Aarhus Universitet, Århus, Denmark 
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Masaki Kashiwara; Niels Lauritzen. Local cohomology and $ \mathcal{D}$-affinity in positive characteristic. Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 993-996. doi : 10.1016/S1631-073X(02)02616-X. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02616-X/

[1] A. Beilinson; J. Bernstein Localisation de 𝔤-modules, C. R. Acad. Sci. Paris, Volume 292 (1981), pp. 15-18

[2] R. Bezrukavnikov, I. Mirković, D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, Preprint, | arXiv

[3] C. Peskine; L. Szpiro Dimension projective finie et cohomologie locale, Inst. Hautes Études Sci. Publ. Math., Volume 42 (1973), pp. 47-119

[4] B. Haastert Über Differentialoperatoren und 𝒟-Moduln in positiver Characteristik, Manuscripta Math., Volume 58 (1987), pp. 385-415

[5] C. Huneke; G. Lyubeznik On the vanishing of local cohomology modules, Invent. Math., Volume 102 (1990), pp. 73-95

[6] G. Kempf The Grothendieck–Cousin complex of an induced representation, Adv. Math., Volume 29 (1978), pp. 310-396

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