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

Schwartz, Lionel¹

¹Université Paris-Nord, Institut Galilée - LAGA, UMR 7539 du CNRS, Av. J.-B. Clément, 93430, Villetaneuse, France

Algebraic and Geometric Topology, Tome 1 (2001) no. 1, pp. 519-548

Résumé

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].

Détail

DOI : 10.2140/agt.2001.1.519

Keywords: Steenrod operations, nilpotent modules, Eilenberg–Moore spectral sequence

Bibliographie

[1] P Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962) 323

[2] H W Henn, J Lannes, L Schwartz, The categories of unstable modules and unstable algebras over the Steenrod algebra modulo nilpotent objects, Amer. J. Math. 115 (1993) 1053

[3] N J Kuhn, On topologically realizing modules over the Steenrod algebra, Ann. of Math. $(2)$ 141 (1995) 321

[4] J Lannes, L Schwartz, À propos de conjectures de Serre et Sullivan, Invent. Math. 83 (1986) 593

[5] D L Rector, Steenrod operations in the Eilenberg–Moore spectral sequence, Comment. Math. Helv. 45 (1970) 540

[6] L Schwartz, La filtration nilpotente de la categorie $\mathcal{U}$ et la cohomologie des espaces de lacets, from: "Algebraic topology—rational homotopy (Louvain-la–Neuve, 1986)", Lecture Notes in Math. 1318, Springer (1988) 208

[7] L Schwartz, Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture, Chicago Lectures in Mathematics, University of Chicago Press (1994)

[8] L Schwartz, À propos de la conjecture de non-réalisation due à N. Kuhn, Invent. Math. 134 (1998) 211

[9] L Schwartz, Unstable modules over the Steenrod algebra, functors, and the cohomology of spaces, from: "Infinite length modules (Bielefeld, 1998)", Trends Math., Birkhäuser (2000) 229

[10] L Smith, On the Künneth theorem I: The Eilenberg–Moore spectral sequence, Math. Z. 116 (1970) 94

[11] L Smith, Lectures on the Eilenberg–Moore spectral sequence, Lecture Notes in Mathematics 134, Springer (1970)