Relating Postnikov pieces with the Krull filtration: A spin-off 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. 
Natalia Castellana, Juan A. Crespo, Jerome Scherer