${\scr D}$-modules arithmétiques. I. Opérateurs différentiels de niveau fini

Berthelot, Pièrre

doi:10.24033/asens.1739

Cited by 127 publications

(308 citation statements)

References 13 publications

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“…Conside Ârons l'exemple donne Â par Berthelot a Á la fin de [Ber96b] : On prend Z p , X b P 1 , U 0 G mk . Soit a P Z (p) un nombre de Liouville, i.e.…”

Section: (Holonomie Sur Y Kunclassified

“…Berthelot a construit le faisceau sur X des ope Ârateurs diffe Ârentiels de niveau fini et d'ordre infini note Â h y X;Q ; ce dernier correspondant a Á la tensorisation par Q (indique Â par l'indice Q) du comple Âte Â faible p-adique (indique Â par le symbole «y») du faisceau classique h X des ope Ârateurs diffe Ârentiels sur X. Enfin, en ajoutant des singularite Âs surconvergentes le long de Z 0 , il construit le faisceau h y X ( y Z 0 ) Q sur X (voir [Ber96b] ou [Ber02] holonome. Cette conjecture a e Âte Â valide Âe dans le cas ou Á X 0 est une courbe propre (voir [Car06b]) puis une courbe (voir [CT08]) et enfin une varie Âte Â projective (voir [Car11c]).…”

Section: Introductionunclassified

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Stabilité de l'holonomie sans structure de Frobenius: cas des courbes

Caro¹

2011

Rend. Sem. Mat. Univ. Padova

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“…Conside Ârons l'exemple donne Â par Berthelot a Á la fin de [Ber96b] : On prend Z p , X b P 1 , U 0 G mk . Soit a P Z (p) un nombre de Liouville, i.e.…”

Section: (Holonomie Sur Y Kunclassified

Section: Introductionunclassified

Stabilité de l'holonomie sans structure de Frobenius: cas des courbes

Caro¹

2011

Rend. Sem. Mat. Univ. Padova

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“…1.3]), D-modules in mixed characteristic can be controlled by certain "Frobenius divisions" (see Definition 10). This important observation, in the present context, is due to Matzat and van der Put [27], [26]; in a more general context it is an application of Berthelot's robust theory of Frobenius descent and arithmetic D-modules [3], [4] and [5].…”

Section: Introductionmentioning

confidence: 99%

“…This result is extremely useful due to the fact that it allows us to translate the problem of dealing with an infinite number of differential operators into a question of commutative algebra (this method is crowned by the elegant solution of the "connected" inverse problem in [27]). After Berthelot's seminal work on arithmetic D-modules and "Frobenius descent" [3], [4], [5], it becomes clear that the generalization of Katz's theorem should also hold. Nevertheless, the precise statement of the desired generalization, which is Theorem 11, is not directly available from [3], [4], [5], but is the subject of [6].…”

Section: φ-Divided and Stratified Sheaves (Applications Of Berthelot'mentioning

confidence: 99%

Lifting $D$-modules from positive to zero characteristic

Santos¹

2011

Bul. Soc. Math. France

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Abstract. -We study liftings or deformations of D-modules (D is the ring of differential operators from EGA IV) from positive characteristic to characteristic zero using ideas of Matzat and Berthelot's theory of arithmetic D-modules. We pay special attention to the growth of the differential Galois group of the liftings. We also apply formal deformation theory (following Schlessinger and Mazur) to analyze the space of all liftings of a given D-module in positive characteristic. At the end we compare the problems of deforming a D-module with the problem of deforming a representation of a naturally associated group scheme.Résumé (Relèvement de D-modules de caractéristique positive en caractéristique nulle)Nous étudions des relèvements des D-modules (D est l'anneau des opérateurs diffé-rentiels de EGA IV) de la caractéristique positive en caractéristique nulle en utilisant des idées de Matzat et la théorie de descente par Frobenius (pour les D-modules arithmétiques) de Berthelot. Nous prêtons une attention particulière à la croissance du groupe de Galois différentiel du relèvement. Nous appliquons aussi la théorie locale des déformations (d'après Schlessinger et Mazur) pour analyser l'espace local de modules des relèvements. À la fin, nous comparons la théorie des déformations (locales) d'un D-module avec la théorie des déformations (locales) d'une représentation d'un schéma en groupes naturellement associé.Texte reçu le 25 mai

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“…It follows that p G * I Gm (1) ∩ ker δ ⊂ I inv Gm (1). Since G is a commutative group scheme, the diagrams (2). By functoriality, the diagram…”

Section: The Morphism ν(•) Gives Also Isomorphisms Of Filtered Complexesmentioning

confidence: 99%

The Filtered Poincaré Lemma in Higher Level (With Applications to Algebraic Groups)

Stum¹,

Quirós²

2008

Nagoya Mathematical Journal

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Abstract. We show that the Poincaré lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincaré lemma for transversal crystals of level m. We interpret the de Rham complex in terms of what we call the Berthelot-Lieberman construction and show how the same construction can be used to study the conormal complex and invariant differential forms of higher level for a group scheme. Bringing together both instances of the construction, we show that crystalline extensions of transversal crystals by algebraic groups can be computed by reduction to the filtered de Rham complexes. Our theory does not ignore torsion and, unlike in the classical case (m = 0), not all invariant forms are closed. Therefore, close invariant differential forms of level m provide new invariants and we exhibit some examples as applications.

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${\scr D}$-modules arithmétiques. I. Opérateurs différentiels de niveau fini

Cited by 127 publications

References 13 publications

Stabilité de l'holonomie sans structure de Frobenius: cas des courbes

Stabilité de l'holonomie sans structure de Frobenius: cas des courbes

Lifting $D$-modules from positive to zero characteristic

The Filtered Poincaré Lemma in Higher Level (With Applications to Algebraic Groups)

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