## Nagoya Mathematical Journal

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

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 17 September 2008

https://projecteuclid.org/euclid.nmj/1221656782

Mathematical Reviews number (MathSciNet)
MR2451221

Zentralblatt MATH identifier
1184.14032

#### Citation

Le Stum, Bernard; Quirós, Adolfo. The filtered Poincaré lemma in higher level (with applications to algebraic groups). Nagoya Math. J. 191 (2008), 79--110. https://projecteuclid.org/euclid.nmj/1221656782

#### References

• P. Berthelot, Cohomologie crystalline des schémas de caractéristique $p>0$, Lecture Notes in Mathematics, vol. 407, Springer-Verlag, Berlin, 1974.
• P. Berthelot, $\mathcalD$-modules arithmétiques I. Opérateurs différentiels de niveau fini, Ann. Sci. École Norm. Sup. (4), 29 (1996), 185--272.
• P. Berthelot, L. Breen and W. Messing, Théorie de Dieudonné crystalline II, Lecture Notes in Mathematics, vol. 930, Springer-Verlag, Berlin, 1982.
• P. Berthelot and A. Ogus, Notes on crystalline cohomology, Princeton University Press, 1978.
• R. Crew, Crystalline cohomology of singular varieties, Geometric aspects of Dwork theory. Vol. I, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 451--462.
• P. Deligne, Théorie de Hodge II, Inst. Hautes Études Sci. Publ. Math., (40) (1971), 5--57.
• J.-Y. Étesse and B. Le Stum, Fonctions $L$ associées aux $F$-isocristaux surconvergents. II. Zéros et pôles unités, Invent. Math., 127 (1997), 1--31.
• L. Illusie, Complexe cotangent et déformations I, Lecture Notes in Mathematics, vol. 239, Springer-Verlag, Berlin, 1971.
• B. Le Stum and A. Quirós, Transversal crystals of finite level, Ann. Inst. Fourier, 47 (1997), 69--100.
• B. Le Stum and A. Quirós, The exact Poincaré lemma in crystalline cohomology of higher level, J. Algebra, 240 (2001), 559--588.
• A. Ogus, $F$-crystals, Griffiths transversality, and the Hodge decomposition, Astérisque, 221 (1994).
• J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986.
• F. Trihan, Fonction $L$ unité d'un groupe de Barsotti-Tate, Manuscripta Math., 96 (1998), 397--419.