On the generalized Novikov first Ext group modulo a prime
Let \(T(m)\) be the Ravenel spectrum at the prime \(p\) characterized by the \(BP_*\)-homology\break \(BP_*(T(m))= \mathbb{Z}_{(p)} [t_1,\dots,t_m] \subset BP_*BP=\mathbb{Z}_{(p)} [t_1,\dots]\). The main result is the structure of the 0-th Ext group \(\text{Ext}^0_{\Gamma (m+1)}(BP_*,M^1_1)\) over the Hopf algebroid \(\Gamma(m+1)= BP_*BP/(t_1,\dots,t_m)\). Here \(M^1_1=v_1^{-1}BP_*/ (p, v_1^\infty)\). The Ext group is computed by \textit{D. C. Ravenel}'s result on \(\text{Ext}^*_{\Gamma(m+1)} (BP_*,v_2^{-1}BP_*/(p))\) [\textit{D. C. Ravenel}, Complex cobordism and stable homotopy groups of spheres, Pure Appl. Math., Academic Press Vol. 121 (1986; Zbl 0608.55001)] and the \(v_1\)-Bockstein spectral sequence associated to the short exact sequence \(v_2^{-1}BP_*/(p) @>1/v_1>> M^1_1@>v_1>> M^1_1\). The group is determined in the same way but independently by \textit{Y. Kamiya} and \textit{K. Shimomura} [Hiroshima Math. J. 31, No. 3, 391-408 (2001; Zbl 1004.55007)], in which \(\text{Ext}^s_{\Gamma(m+1)} (BP_*,M^1_1)\) is determined for all \(s\geq 0\) and applied to determine the homotopy groups \(\pi_* (L_2V(0) \wedge T(m))\), where \(V(0)\) denotes the mod \(p\) Moore spectrum. In this paper, the authors apply the result to determine the first line \(\text{Ext}^1_{ \Gamma(m+1)} (BP_*,BP_*/(p))\) of the \(E_2\)-term of the Adams-Novikov spectral sequence converging to \(\pi_*(V(0)\wedge T(m))\), and wrote down the first line in the same form as \textit{H. R. Miller} and \textit{W. S. Wilson}'s result on \(\text{Ext}^1_{BP_*BP} (BP_*, BP_*/(p))\) for \(\pi_*(V(0))\) [Topology 15, 131-141 (1976; Zbl 0337.55006)]. These are seen immediately from the structures of \(\text{Ext}^0_{\Gamma(m+1)} (BP_*,M^1_1)\) for \(m\geq 0\).
scientific article; zbMATH DE number 1874589