Volume 1, issue 1 (2001)

Download this article
For printing
Recent Issues

Volume 19
Issue 7, 3217–3753
Issue 6, 2677–3215
Issue 5, 2151–2676
Issue 4, 1619–2150
Issue 3, 1079–1618
Issue 2, 533–1078
Issue 1, 1–532

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

Author Index
The Journal
About the Journal
Editorial Board
Subscriptions
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
To Appear
 
Other MSP Journals
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

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 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].

Keywords
Steenrod operations, nilpotent modules, Eilenberg–Moore spectral sequence
Mathematical Subject Classification 2000
Primary: 55S10
Secondary: 57S35
References
Forward citations
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