The first line of the Bockstein spectral sequence on a monochromatic spectrum at an odd prime (Q2911019)

Let \(BP\) be the Brown-Peterson spectrum for an odd prime \(p\). The authors are interested in understanding NEWLINE\[NEWLINE\pi_\ast(S^0_{(p)}) \Leftarrow \mathrm{Ext}_{BP_\ast BP}^{\ast,\ast}(BP_\ast, BP_\ast) \Leftarrow E_1^{s,t}(n-s) \Leftarrow E_1^{s-1,t+1}(n-s+1),NEWLINE\]NEWLINE a diagram of spectral sequences that (going from left to right) consists of an Adams-Novikov spectral sequence, the chromatic spectral sequence, and a Bockstein spectral sequence. The sources of the above arrows are an \(E_2\)-term of Ext groups for comodules, an \(E_1\)-term, and another \(E_1\)-term, respectively, with NEWLINE\[NEWLINEE_1^{s,t}(n-s) = \mathrm{Ext}^t_{E(n)_\ast E(n)}(E(n)_\ast, M^s_{n-s}),NEWLINE\]NEWLINE where \(E(n)\) is equal to the Johnson-Wilson spectrum.NEWLINENEWLINEThe term \(E_1^{s,t}(n-s)\) is currently known in only a small number of cases and much of this knowledge is due to the second author of the paper under review, together with his collaborators. Perhaps the most influential paper in the second author's oeuvre on these computations is [\textit{K. Shimomura} and \textit{A. Yabe}, Topology 34, No. 2, 261--289 (1995; Zbl 0832.55011)]. In the paper under review, the authors compute the structure of \(E_1^{1,1}(n-1)\) when \(n > 3\), by using the above Bockstein spectral sequence. In more detail, they show that the Ext group \(E_1^{1,1}(n-1)\) is a direct sum of eight modules and, for example, the fourth summand in this direct sum is an infinite direct sum. The basic statistics in the preceding sentence were given to illustrate that this paper involves impressive computational work.NEWLINENEWLINENow let \(p > 5\) and let \(\gamma\) be the self-map on \(V(2)\) due to Toda. As an application of the above work, the authors show that the elements \(\gamma^t i \alpha_1\) and \(\gamma^t i \beta_1\) are nontrivial in \(\pi_\ast(V(2))\) for \(t > 0\).