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

Abstract

The chromatic spectral sequence is introduced in \cite{mrw} to compute the $E_2$-term of the \ANSS\ for computing the stable homotopy groups of spheres. The $E_1$-term $E_1^{s,t}(k)$ of the spectral sequence is an Ext group of $BP_*BP$-comodules. There are a sequence of Ext groups $E_1^{s,t}(n-s)$ for non-negative integers $n$ with $E_1^{s,t}(0)=E_1^{s,t}$, and Bockstein spectral sequences computing a module $E_1^{s,*}(n-s)$ from $E_1^{s-1,*}(n-s+1)$. So far, a small number of the $E_1$-terms are determined. Here, we determine the $E_1^{1,1}(n-1)=\e^1M^1_{n-1}$ for $p>2$ and $n>3$ by computing the Bockstein spectral sequence with $E_1$-term $E_1^{0,s}(n)$ for $s=1,2$. As an application, we study the non-triviality of the action of $\alpha_1$ and $\beta_1$ in the homotopy groups of the second Smith-Toda spectrum V(2).