La filtration de Krull de la catégorie 𝒰 et la cohomologie des espaces

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

Keywords

Steenrod operations, nilpotent modules, Eilenberg–Moore spectral sequence

Mathematical Subject Classification 2000

Primary: 55S10

Secondary: 57S35

References

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Publication

Received: 9 October 2000

Revised: 4 July 2001

Accepted: 30 September 2001

Published: 5 October 2001

Authors

Lionel Schwartz
Université Paris-Nord Institut Galilée - LAGA UMR 7539 du CNRS Av. J.-B. Clément 93430, Villetaneuse France

Volume 1, issue 1 (2001)

Lionel Schwartz