The \(\text{Ext}^0\)-term of the real-oriented Adams-Novikov spectral sequence (Q2752873)

In an earlier paper [Topology 40, No. 2, 317-399 (2001; Zbl 0967.55010)] the author together with \textit{I. Kriz} introduced the Real-oriented Adams-Novikov spectral sequence, an Adams type spectral sequence based on the real oriented homology theory \(BP{\mathbb R}_{\star}\)() of Araki and Landweber, which converges to \(\pi_{\star}(^{{\mathbb Z}/2} S^0)^{\wedge}_2\) the \({\mathbb Z}/2\) equivariant stable homotopy of the sphere spectrum completed at \(2\). Although the Araki/Landweber spectrum was originally constructed as a bigraded spectrum, it is now described as a \(RO({\mathbb Z}/2)\) graded spectrum, thus \(\star = k + \ell \alpha\) where \(k, \ell \in {\mathbb Z}\) and \(\alpha\) is the sign representation. The \(E_2\) term of this spectral sequence is \(\text{Ext}^*_{BP{\mathbb R}_\star BP{\mathbb R}}(BP{\mathbb R}_\star, BP{\mathbb R}_\star)\) and the paper under review is devoted to describing the \(0\) line of this \(E_2\) term. The main result is that \(\text{Ext}^0\) is equal to NEWLINE\[NEWLINE{\mathbb Z}_{(2)} \{ v_0 \sigma^{2 \ell} \mid \ell \in {\mathbb Z} \} \oplus a \cdot {\mathbb Z}/ 2 [a] \oplus {\mathbb Z}/2 \{ v^r_n \sigma^{\ell \cdot 2^{n+1}} a^t \mid n,\;r\geq 1,\;\ell \in {\mathbb Z},\;2^n -1 \leq t \leq 2^{n+1} -2\}NEWLINE\]NEWLINE where \(v_{n}\), \(a\) and \(\sigma\) have degrees \((2^n-1)(1+\alpha), -\alpha\) and \(\alpha -1\) respectively. In addition to proving this result the author also makes the case for considering the elements \(v_n \sigma^{\ell 2^{n+1}} a^{2^n-1}\) as elements of Hopf invariant one.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00013].