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

TL;DR

This paper computes specific Ext groups in the chromatic spectral sequence at an odd prime, advancing understanding of the Bockstein spectral sequence and its implications for stable homotopy groups.

Contribution

It determines new E_1-term values in the Bockstein spectral sequence for p>2 and n>3, providing explicit calculations previously unavailable.

Findings

Computed E_1^{1,1}(n-1) for p>2 and n>3

Analyzed the action of  and  on homotopy groups of V(2)

Extended the understanding of the chromatic spectral sequence structure

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).