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

Abstract

The chromatic spectral sequence was introduced by Miller, Ravenel, and Wilson to compute the $E_2$-term of the Adams-Novikov spectral sequence for computing the stable homotopy groups of spheres. The $E_1$-term $E_1^{s,t}\left(k\right)$ of the spectral sequence is an Ext group of $B{P}_{\ast }BP$-comodules. There is a sequence of Ext groups $E_1^{s,t}(n-s)$ for nonnegative integers $n$ with $E_1^{s,t}\left(0\right)=E_1^{s,t}$, and there are Bockstein spectral sequences computing a module $E_1^{s,\ast }(n-s)$ from $E_1^{s-1,\ast }(n-s+1)$. So far, a small number of the $E_1$-terms are determined. Here, we determine the $E_1^{1,1}(n-1)={Ext}^1{M}_{n-1}^1$ for $p>2$ and $n>3$ by computing the Bockstein spectral sequence with $E_1$-term $E_1^{0,s}\left(n\right)$ for $s=1,2$. As an application, we study the nontriviality of the action of ${\alpha }_1$ and ${\beta }_1$ in the homotopy groups of the second Smith-Toda spectrum $V\left(2\right)$.