Algebraic & Geometric Topology 3 (2003) 399–433

DOI: 10.2140/agt.2003.3.399

arXiv: math.AT/0306271

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

Mathematical Subject Classification

Primary: 55S10

Secondary: 55T20, 57T35

References

Forward citations

Publication

Received: 29 November 2002
Revised: 3 May 2003
Accepted: 14 January 2003
Published: 8 May 2003

Authors