On the Adams-Novikov spectral sequence and products of \(\beta\)-elements
Let \(p\geq 5\) be a prime, BP the Brown-Peterson spectrum at p, and \((A,\Gamma)=(BP_*,BP_*BP)\) the Hopf-algebroid corresponding to BP. With common notation, \(A={\mathbb{Z}}_{(p)}[v_ 1,...,v_ n,..]\) and \(\Gamma =A[t_ 1,...,t_ n,...]\). \textit{H. R. Miller}, \textit{D. C. Ravenel} and \textit{W. S. Wilson} [Ann. Math., II. Ser. 106, 469-516 (1977; Zbl 0374.55022)] studied the Adams-Novikov spectral sequence \(Ext^*_{\Gamma}(A,A) \Rightarrow \pi_*(S^ 0)\) by means of a chromatic spectral sequence \(Ext^*_{\Gamma}(A,M^*_ n) \Rightarrow Ext^*_{\Gamma}(A,N^ 0_ n)\); here \(N^ 0_ n=A/(p,v_ 1,...,v_{n-1})\) and, inductively, \(M^ s_ n=v^{-1}_{n+s} N^ s_ n\) with short exact sequences \(N^ s_ n\to M^ s_ n\to N_ n^{s+1}.\) In the paper under review the groups \(Ext^ s_{\Gamma}(A,M^ 1_ 1)\) are computed completely, using a change of rings argument and Ext- calculations over \(B={\mathbb{Z}}_{(p)}[v_ 1,v_ 2,v_ 2^{-1}]\). It turns out that the Adams-Novikov spectral sequence starting with \(E_ 2=Ext^*_{\Gamma}(A,M^ 1_ 1)\) must be trivial. As an application the author obtains new results about the vanishing or nonvanishing of products \(\beta_ s\cdot \beta_{tp/j}\) and \(\beta_{sp/i}\cdot \beta_{tp/j}\) in \(\pi_*(S^ 0)\).
scientific article; zbMATH DE number 3977930