  Paper written in French -- English abstract:
  This paper proves a particular case of a conjecture of N. Kuhn. This
conjecture is as follows. Consider the Gabriel-Krull filtration of the category
U of unstable modules.
  Let U_n, n>=0, be the n-th step of this filtration. The category U is the
smallest thick sub-category that contains all sub-categories U_n and is stable
under colimit [L. Schwartz, Unstable modules over the Steenrod algebra and
Sullivan's fixed point set conjecture, Chicago Lectures in Mathematics Series
(1994)]. The category U_0 is the one of locally finite modules, i.e. the
modules that are direct limit of finite modules. The conjecture is as follows,
let X be a space then :
  * either H^*X is locally finite, * or H^*X does not belong to U_n, for all n.
  As an example the cohomology of a finite space, or of the loop space of a
finite space are always locally finite. On the other side the cohomology of the
classifying space of a finite group whose order is divisible by 2 does belong
to any sub-category U_n. One proves this conjecture, modulo the additional
hypothesis that all quotients of the nilpotent filtration are finitely
generated. This condition is used when applying N. Kuhn's reduction of the
problem. It is necessary to do it to be allowed to apply Lannes' theorem on the
cohomology of mapping spaces.[N. Kuhn, On topologically realizing modules over
the Steenrod algebra, Ann. of Math. 141 (1995) 321-347].
