Nagoya Mathematical Journal

The filtered Poincaré lemma in higher level (with applications to algebraic groups)

Bernard Le Stum and Adolfo Quirós

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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.

Article information

Nagoya Math. J., Volume 191 (2008), 79-110.

First available in Project Euclid: 17 September 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F30: $p$-adic cohomology, crystalline cohomology
Secondary: 14F40: de Rham cohomology [See also 14C30, 32C35, 32L10] 14L15: Group schemes

crystalline cohomology of higher level transversal crystals filtered de Rham complex group schemes invariant differential forms


Le Stum, Bernard; Quirós, Adolfo. The filtered Poincaré lemma in higher level (with applications to algebraic groups). Nagoya Math. J. 191 (2008), 79--110.

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