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In this paper we develop techniques to compute the cooperations algebra forthe second truncated Brown-Peterson spectrum $\tBP{2}$. We prove that thecooperations 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 basisof the $v_2$-torsion free part.

On $BP\langle 2\rangle$-cooperations