Espaces profinis et problèmes de réalisabilité

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.