On the chromatic $\e^0(M^1_{n-1})$ on $\Ga(m+1)$ for an odd prime

Abstract

Ravenel introduced spectra $T(m)$ for $m\ge 0$ interpolating the Brown-Peterson spectrum $BP$ and the sphere spectrum $S$ in Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, Providence, 2004. Since the homotopy groups of $BP$ are well known, it is interesting to study differences among the homotopy groups of $T(m)$'s to study the homotopy groups of spheres. He also introduced the localization functor $L_n$ on the stable homotopy category in "Localization with respect to certain periodic homology theories," Amer. J. Math. 106 (1984), 351–414. To study the difference of $L_nT(m)$'s for a fixed integer $n$, we consider the corresponding chromatic $E_1$-term $\e^0(M^1_{n-1})$ on $\Ga(m+1)$ for each $m$, and determine it for $m+1\ge (n-2)(n-1)$ in this paper. The results show that the structures depend on a integer $\Lt[\dfrac{m+1}{n-1}\Rt]$. Here $[x]$ denotes the greatest integer that does not exceed $x$.