On \(BP\langle 2\rangle\)-cooperations

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.