An exact sequence related to Adams-Novikov \(E_ 2\)-terms of a cofibering
Let \(L_ n X\) denote the Bousfield localization of a spectrum \(X\) with respect to the Johnson-Wilson spectrum \(E(n)\), and \(\eta_ X: X\to L_ n X\), the localization. Then \textit{D. C. Ravenel} gives the cofibration \(\Sigma^{-n-1} N_{n+1} X\to X@> \eta X>> L_ n X\) in [Am. J. Math. 106, 351-414 (1984; Zbl 0586.55003), Th. 5.10]. Take \(X\) to be the Toda- Smith spectrum \(V(j-1)\). Then the Brown-Peterson homology \(BP_ * (N_{n+1} X)\) of \(N_{n+1} X\) is \(N_ j^{n+1}\) of \textit{H. R. Miller}, \textit{D. C. Ravenel} and \textit{W. S. Wilson}'s sense [Ann Math., II. Ser. 106, 469-516 (1977; Zbl 0374.55022)]. The \(E_ 2\)-terms of the Adams-Novikov spectral sequences converging to \(\pi_ * (N_{n+1} X)\), \(\pi_ * (X)\) and \(\pi_ * (L_ n X)\) are \(\text{Ext}_ \Gamma (A, N_ j^{n+1-j})\), \(\text{Ext}_ \Gamma (A, A/I_ j)\) and \(\text{Ext}_ \Sigma (B, B/I_ j)\), respectively, for the Hopf algebroids \((A, \Gamma)= (BP_ *, BP_ * (BP))\) and \((B, \Sigma)= (E(n)_ *, E(n)_ * (E(n)))\), and for the ideal \(I_ j=(p, v_ 1,\dots, v_{j-1})\). One of the results is the long exact sequence \[ \begin{multlined} \cdots\to\text{Ext}_ \Gamma^ t (A,N_ j^{n+1-j}) \overset \eta{\rightarrow} \text{Ext}_ \Gamma^{t+ n+1} (A,A/I_ j) \overset(\eta_ X)_ {*} {\longrightarrow} \text{Ext}_{\Sigma'}^{t+n +1} (B, B/I_ j)\\ \text{Ext}_ \Gamma^{t+1} (A, N_ j^{n+ 1-j}{)} \cdots ,\end{multlined} \] where \(\eta\) denotes the universal Greek letter map and \((\eta_ X )_ *\), the induced map from the localization. Here the universal Greek letter map is introduced by Miller, Ravenel and Wilson [loc. cit.]. This follows from the main result which claims the long exact sequence involving the universal Greek letter map and is proved algebraically. The result on \(V(j)\) is generalized to spectra \(W_ k(i)\) characterized by \(BP_ * (W_ k (i))= BP_ */ I_{i+1} [t_ 1, \dots, t_ k]\).
scientific article; zbMATH DE number 758150