An algebraic generalization of image \(J\) (Q1004515)

D. C. Ravenel introduced a method called `infinite descent' in order to compute the homotopy groups \(\pi_*(S^0)\) of spheres. In the method, he introduced the spectra \(T(m)\) and \(T(m)_{(j)}\), whose \(BP_*\)-homologies are \(BP_*[t_1,\dots,t_m]\subset BP_*BP\) and a free \(BP_*(T(m))\)-module over generators \(t_{m+1}^\ell\) for \(0\leq \ell<p^j\), respectively. Besides, he also developed a method for computing \(\operatorname{Im}J\) and \(\operatorname{Coker}J\) of \(\pi_*(S^0)\) by use of the Adams-Novikov spectral sequence. In particular, this method works to determine the first line of the \(E_2\)-term, which is \(\operatorname{Im}J\) for an odd prime. In this paper, the author applies this method to study the first line of the Ext group Ext\(_{BP_*(BP)}^1(BP_*,BP_*(T(m))_{(j)})\), and determines it for the cases \(j=1\) at any prime, and \(j=2\) at the prime 2. Note that this is \(\operatorname{Im}J\) if \(m\) and \(j\) are both zero.