The chromatic \(E_1\)-term \(H^0 M_1^2\) for \(p>3\) (Q1965914)

The chromatic spectral sequence was introduced by Miller, Ravenel and Wilson to compute the \(E_2\)-term \(E^*_2(V(n- 1))\) of the Adams-Novikov spectral sequence converging to the homotopy groups \(\pi_*(V(n))\) at the prime \(p\). Here \(V(n)\) denotes the Smith-Toda spectrum, and is known to exist for \(n\leq 3\) and \(p>2n\). The \(E_1\)-term of the chromatic spectral sequence is written as \(H^t H^s_n\), which is studied when \(s= 0\), \(s+ n\leq 2\) or \((s,t)= (1,0)\). In this paper, the module structure of \(H^0 M^2_1\) is determined at a prime \(>3\). The result and the proof show how complicated the structure is. The proof is analogous to the determination of \(H^0 M^2_0\) by Miller, Ravenel and Wilson, while the computation is much harder. This result is also related to the homotopy groups \(\pi_*(L_3 V(0))\) and \(\pi_*(V(0))\) by the Adams-Novikov spectral sequence, where \(L_3\) denotes the Bousfield localization functor with respect to \(v^{-1}_3 BP\).