This paper proves a particular case of a conjecture of
N Kuhn. Consider the Gabriel–Krull filtration of the category U of
unstable modules. Let Un, n ≥ 0, denote the nth
term of this filtration. The category U is the smallest thick subcategory
that contains all subcategories Un and is stable under colimit
[L Schwartz, Unstable modules over the Steenrod algebra and Sullivan's
fixed point set conjecture, Chicago Lectures in Mathematics (1994)].
The category U0 is the category of locally finite modules,
that is, the modules that direct limits of finite modules.
The conjecture is as follows: Let X be a space, then
either H*X is locally finite,
or H*X not in Un, for all n.
For example, the cohomology of a finite-dimensional space, or that of
the loop space of a finite space are always in U0. The
cohomology of the classifying space of a finite group of even order does
not belong to any subcategory Un.
We prove this conjecture, modulo the additional hypothesis that all
quotients of the nilpotent fibration are finitely generated. This
condition implies in particular that the cohomology is finite-dimensional
in each degree. This is necessary to apply Lannes' theorem on the
cohomology of mapping spaces, which is required for N Kuhn's reduction of
the problem [N Kuhn, On topologically realizing modules over the
Steenrod algebra, Ann. of Math. 141 (1995) 321–347].
For simplicity, this article does not consider the