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

Lionel Schwartz

Algebraic & Geometric Topology 1 (2001) 519–548

arXiv: math.AT/0110230


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 Un, for n ≄ 0, be the nth step 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 Series (1994)]. The category U0 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∗X ∈U0, or H∗X∉Un, 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 Un. 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].

Steenrod operations, nilpotent modules, Eilenberg–Moore spectral sequence
Mathematical Subject Classification 2000
Primary: 55S10
Secondary: 57S35
Forward citations
Received: 9 October 2000
Revised: 4 July 2001
Accepted: 30 September 2001
Published: 5 October 2001
Lionel Schwartz
Université Paris-Nord
Institut Galilée - LAGA
UMR 7539 du CNRS
Av. J.-B. Clément
93430, Villetaneuse