*Séminaire de Géométrie Algébrique de Rennes -
Exposé du
18 mai 2006
*

* *

**Adolfo Quirós** (Universitad Autonoma de Madrid)

**Titre** :
Differential forms
of higher levels and
algebraic groups.

**Résumé** :

Using a "Berthelot-Lieberman complex", we can prove a filtered Poincaré lemma that allows us to calculate crystalline cohomology of higher level
without ignoring torsion, which is in fact very rich in this theory.

The same Berthelot-Lieberman construction can be used to define the conormal complex of higher level of a group scheme and to study its relation
to invariant differential forms of higher level. Unlike the classical case (level 0), not all invariant forms are closed. Actually, the module of
closed invariant forms is isomorphic to the first cohomology group of the conormal complex. We will present concrete examples, including the
Legendre family of elliptic curves, and give the relation with de Rham cohomology of higher level in the case of abelian schemes.

Putting together these two instances of the Berthelot-Lieberman construction, we can compute crystalline extensions of transversal crystals by
algebraic groups by reduction to filtered de Rham complexes.

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