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

Published online by Cambridge University Press:  11 January 2016

Bernard Le Stum  and
Adolfo Quirós
Bernard Le Stum
Affiliation:
IRMAR, Université de Rennes I, Campus de Beaulieu, F-35042 Rennes Cedex, France, bernard.le-stum@univ-rennes1.fr
Adolfo Quirós
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, E-28049 Madrid, Spain,adolfo.quiros@uam.es
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Abstract

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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.

Keywords

14F3014F4014L15
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 191 , 2008 , pp. 79 - 110
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2008

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