#### Algebraic and Geometric Topology 3 (2003),
paper no. 13, pages 399-433.

## Espaces profinis et problemes de realisabilite

### Francois-Xavier Dehon and Gerald Gaudens

**Abstract**.
The mod p cohomology of a space comes with an action of the Steenrod
Algebra. L. Schwartz [A propos de la conjecture de non realisation due
a N. Kuhn, Invent. Math. 134, No 1, (1998) 211--227] proved a
conjecture due to N. Kuhn [On topologicaly realizing modules over the
Steenrod algebra, Annals of Mathematics, 141 (1995) 321--347] stating
that if the mod $p$ cohomology of a space is in a finite stage of the
Krull filtration of the category of unstable modules over the Steenrod
algebra then it is locally finite. Nevertheless his proof involves
some finiteness hypotheses. We show how one can remove those
finiteness hypotheses by using the homotopy theory of profinite spaces
introduced by F. Morel [Ensembles profinis simpliciaux et
interpretation geometrique du foncteur T, Bull. Soc. Math. France, 124
(1996) 347--373], thus obtaining a complete proof of the
conjecture. For that purpose we build the Eilenberg-Moore spectral
sequence and show its convergence in the profinite setting.
**Keywords**.
Steenrod operations, nilpotent modules, realization, Eilenberg-Moore spectral sequence, profinite spaces

**AMS subject classification**.
Primary: 55S10.
Secondary: 55T20, 57T35.

**DOI:** 10.2140/agt.2003.3.399

**E-print:** `arXiv:math.AT/0306271`

Submitted: 29 November 2002.
(Revised: 3 May 2003.)
Accepted: 14 January 2003.
Published: 8 May 2003.

Notes on file formats
Francois-Xavier Dehon and Gerald Gaudens

Laboratoire J.A. Dieudonne, Universite de Nice Sophia-Antipolis

Parc Valrose - BP 2053 - 06101 Nice, France

and

Laboratoire Jean Leray (UMR 6629 du C.N.R.S.), Universite de Nantes

BP 92208 - 44322 Nantes Cedex 3, France

Email: dehon@math.unice.fr, gaudens@math.univ-nantes.fr

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