Relating Postnikov pieces with the Krull filtration: a spin-o¤ of Serre’s theorem

We characterize H-spaces which are p-torsion Postnikov pieces of finite type by a cohomological property together with a necessary acyclicity condition. When the mod p cohomology of an H-space is finitely generated as an algebra over the Steenrod algebra we prove that its homotopy groups behave like those of a finite complex. In particular, a p-complete infinite loop space has a finite number of non-trivial homotopy groups if and only if its mod p cohomology satisfies this finiteness condition. 2000 Mathematics Subject Classification: 55P45; 55P20, 55P60, 55P47.