## Algebraic & Geometric Topology

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

Lionel Schwartz

#### 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 $Un$, for $n≥0$, be the $n$th 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].

#### Article information

Source
Algebr. Geom. Topol., Volume 1, Number 1 (2001), 519-548.

Dates
Revised: 4 July 2001
Accepted: 30 September 2001
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882607

Digital Object Identifier
doi:10.2140/agt.2001.1.519

Mathematical Reviews number (MathSciNet)
MR1875606

Zentralblatt MATH identifier
1007.55014

Subjects
Primary: 55S10: Steenrod algebra
Secondary: 57S35

#### Citation

Schwartz, Lionel. La filtration de Krull de la catégorie $\mathcal{U}$ et la cohomologie des espaces. Algebr. Geom. Topol. 1 (2001), no. 1, 519--548. doi:10.2140/agt.2001.1.519. https://projecteuclid.org/euclid.agt/1513882607

#### References

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