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Abstract
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Dans cet article, on introduit et on étudie le concept de
-localisation qui est la variante
du concept de
-localisation
où l’on remplace la droite affine par la droite projective. On démontre un théorème de
-connexité
en adaptant la preuve de Morel de son théorème de
-connexité. On s’intéresse
ensuite à la
-localisation
du faisceau des formes différentielles absolues et on montre que son
-ième
faisceau d’homologie est nul. Ceci nous amène naturellement à la définition d’une
classe de Kodaira–Spencer arithmétique. Enfin, nous montrons un lien entre cette
classe de Kodaira–Spencer arithmétique et les classes de Deligne–Illusie pour
presque tout nombre premier, ce qui nous permet de prouver qu’elle est non
identiquement nulle.
In this article, we introduce and study the concept of
-localisation which is the variant
of the concept of
-localisation
where the affine line is replaced by the projective line. We prove a
-connectivity
theorem following the proof of Morel of his
-connectivity theorem. We
then consider the
-localisation
of the sheaf of absolute differential forms and we show that its
-th
homology sheaf vanishes. This naturally brings us to the definition of an
arithmetic Kodaira–Spencer class. Finally, we establish a link between this
arithmetic Kodaira–Spencer class and the Deligne–Illusie classes for almost all
prime numbers, which enables us to prove that the former is not identically
zero.
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Keywords
$\mathbb{P}^1$-localisation, formes différentielles, classe
de Kodaira–Spencer arithmétique
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Mathematical Subject Classification 2010
Primary: 14F05, 14F10, 14F40, 14F42, 14G25
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Milestones
Received: 27 February 2019
Revised: 6 May 2020
Accepted: 21 May 2020
Published: 5 December 2020
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