On the chromatic Ext\(^0(M^1_{n-1})\) on \(\Gamma(m+1)\) for an odd prime
Fix an odd prime number. Let \(T(m)\) for \(m\geq 0\) be the Ravenel spectrum characterized by the Brown-Peterson homology \(BP_*(T(m))=BP_*[t_1,\dots,t_m]\). It is interesting to study the homotopy groups of \(\pi_*(T(m))\), since these spectra interpolate between the sphere spectrum and the Brown-Peterson spectrum. In [Ann. Math. (2) 106, 469--516 (1977; Zbl 0374.55022)], \textit{H.~R.~Miller, D.~C.~Ravenel} and \textit{W.~S.~Wilson} introduced the chromatic spectral sequence converging to the \(E_2\)-term of the Adams-Novikov spectral sequence for computing the homotopy groups of the Smith-Toda spectrum \(V(n)\) with \(E_1^{s,t}=H^tM_{n+1}^s\). Here \(H^*\) denotes the Ext group of \(BP_*BP\)-comodules, and \(V(n)\) is characterized by \(BP_*(V(n))=BP_*/(p,v_1,\dots, v_n)\). There is an analogous spectral sequence converging to the \(E_2\)-term of the Adams-Novikov spectral sequence for computing the homotopy groups of \(T(m)\wedge V(n)\). The \(E_1\)-term is \(H_m^tM^s_{n+1}\) for the Ext group \(H_m^*\) of \(BP_*BP/(t_1,\dots,t_m)\)-comodules. The \(E_1\)-terms \(H_m^tM^0_{n}\) were determined by \textit{D.~C.~Ravenel} [Complex cobordism and stable homotopy groups of spheres. Pure and Applied Mathematics, Vol. 121. Orlando etc.: Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers) (1986; Zbl 0608.55001)]. There is also a Bockstein spectral sequence converging to \(H^*_mM^s_{n-s}\) with \(E_1\)-term \(H^*_mM^{s-1}_{n-s+1}\). By use of these, \(H^*_mM_{n-1}^1\) was determined by the reviewer for \(m+1\geq n(n-1)\) [Contemp. Math. 293, 285--297 (2002; Zbl 1012.55020)]. For \(m+1<n(n-1)\), the \(E_1\)-term \(H^0_mM^1_2\) was determined by \textit{I.~Ichigi, H.~Nakai} and \textit{D.~C.~Ravenel} [Trans. Am. Math. Soc. 354, No. 9, 3789--3813 (2002; Zbl 0996.55019)]. This paper completely describes \(H^0_mM^1_{n-1}\) for \(m+1\geq (n-2)(n-1)\), and shows some formulas on the generators of \(H^0_mM^1_{n-1}\) for \(m+1> n-1\), where the result on \(H_m^tM^0_{n}\) follows from Ravenel's work. The structure of the \(E_1\)-term depends on \(k\) in the inequality \(k(n-1)\leq m+1<(k+1)(n-1)\).
scientific article; zbMATH DE number 5956517