Corollary 13 ([The action of $p$ on $Y$]). For the spectrum Y of Proposition 12 ([Construction through V(3) ]), π_0Y=𝔽_p and pid_Y is nullhomotopic.

The bottom two cells form the mod- p Moore spectrum. All other cells have dimension at least qu_1+1=q+1>1 , so they do not change π_0 . Thus pη_3=0 in π_0Y . Applying the left unit identity of its binary multiplication gives pid_Y =μ_3((pη_3)_Y)=0.