La filtration de Krull de la catégorie $\mathcal{U}$ et la cohomologie des espaces

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 $\mathcal{U}$ of unstable modules.

Let ${\mathcal{U}}_n$, for $n\ge 0$, be the $n$th step of this filtration. The category $\mathcal{U}$ is the smallest thick subcategory that contains all subcategories ${\mathcal{U}}_n$ 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 ${\mathcal{U}}_0$ is the one of locally finite modules, that is, the modules that are direct limits of finite modules. The conjecture is as follows: Let $X$ be a space, then either $H^{\ast }X\in {\mathcal{U}}_0$, or $H^{\ast }X\notin {\mathcal{U}}_n$, for all $n$.

As an examples, 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 subcategory ${\mathcal{U}}_n$. 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].