On $BP\langle 2\rangle$-cooperations

TL;DR

This paper develops methods to compute the algebra of cooperations for the second truncated Brown-Peterson spectrum, revealing its decomposition and providing a recursive basis construction.

Contribution

It introduces techniques for computing the cooperations algebra of P and describes its decomposition and basis construction.

Findings

Decomposition of P}_* into a direct sum with specific properties.

Identification of the algebra as a _2[v_0,v_1,v_2]-module with even degree concentration.

Development of a recursive procedure for basis construction of the torsion-free part.

Abstract

In this paper we develop techniques to compute the cooperations algebra for the second truncated Brown-Peterson spectrum $\tBP{2}$. We prove that the cooperations algebra $\tBP{2}_*\tBP{2}$ decomposes as a direct some of a $\F_2$-vector space concentrated in Adams filtration 0 and a $\F_2[v_0,v_1,v_2]$-module which is concentrated in even degrees and $v_2$-torsion free. A recursive procedure is also developed to provide an basis of the $v_2$-torsion free part.