The structure of the general chromatic \(E_1\)-term Ext\(^0_{\varGamma(2)}\) (BP\(_{\ast}\),M\(^1_2\)) at the prime 2 (Q1409331)

This paper presents an explicit description of the Ext group named in the title. If \(T(m)\) denotes the Ravenel spectrum, whose \(BP\) homology is \(BP_* [t_1 \dots t_m]\), then the Adams Novikov spectral sequence convergent to \(\pi_*(T(m))\) has \(E_2\) term \(\text{Ext}^*_{\Gamma(m+1)} (BP_*, BP_*)\) where \(\Gamma(m+1) = BP_*[t_{m+1}, t_{m+2}, \dots]\). The chromatic spectral sequence convergent to this \(E_2\) term has as its \(E_1\) term \(\text{Ext}^t_{\Gamma(m+1)} (BP_*, M^s_0)\) where \(M^n_m = v^{-1}_{m+n} BP_*/(p, \dots, v_{m-1}, v^\infty_m \dots v^\infty_{m+n-1})\). These Ext groups have all been previously described except for the case \(p=2, m =1\). The authors inductively define integers \(\hat{a}(k)\) and elements \(\hat{x}_k\) and prove that, as a \(v^{-1}_3 \hat{k}(2)_*\) module, \(E^0_{\Gamma(2)} (BP_*, M^1_2)\) is the direct sum of cyclic \({\mathbb Z}/(2) [v_2, v^{\pm 1}_3]\)-modules generated by \(\hat{x}^s_k/v^{\hat{a}(k)}_2\) for \(k \geq 0\), \(s> 0\), \(s\) odd, and \(v^{-1}_3 \hat{K}(2)_*/\hat{k}(2)_*\) generated by \(1/v^j_2\) for \(j \geq 1\). Here \(\hat{K}(2)_* = {\mathbb Z}/(2) [v^{\pm}_2, v_3]\) and \(\hat{k}(2) = {\mathbb Z}/(2) [v_2, v_3]\).