Laurent Piriou
Université de Nantes,
Département de mathématiques
2 rue de la Houssinière
BP 92208 Nantes Cedex 03
France

laurent.piriou@math.univ-nantes.fr

Lionel Schwartz
Université Paris 13
Institut Galilée
LAGA UMR 7539 du CNRS
Av. J. B. Clément
93430 Villetaneuse France
schwartz@math.univ-paris13.fr



Code AMS 55S10

This article considers two filtrations on the mod-$2$ cohomology
$H^*E$
of an abelian $2$-groups $E$. The first one is the primitive fitration, recall
that $H^*E$ is a Hopf algebra.  The second one is a kind
of socle or Loewy filtration of $H^*E$ as unstable module. If
dimension of $E$ is $1$ the two filtrations are the same, if 
the dimension is larger than $2$ it is shown that
the filtration are, in some sense compatible.

There is an analogous statement 
in ${\cal F}$,  the category of functors from the category of finite dimensional
${\bf F}_2$-vector spaces to the category of all
${\bf F}_2$-vector spaces, for the functor $V \mapsto 
{\rm map}({\rm Hom}(V,E),{\bf F}_2)$. 

However, it is better to work with unstable modules because 
the Steenrod algebra allows computation on certain classes, that 
are central in the proof, given by the representation theory of 
symmetric groups that are central in the proof.