A vanishing line in the \(BP\langle 1\rangle\)-Adams spectral sequence (Q1585058)

For an odd prime \(p\) let \(\ell\) denote the bottom indecomposable summand of the \(p\)-local connective \(K\)-theory spectrum (\(\ell= BP\langle 1\rangle\)). Moreover, let \(E_n^{*,*}(X;\ell)\) denote the \(E_n\)-term of the \(\ell\)-Adams-Novikov spectral sequence for a connective finite type spectrum \(X\). In the present paper the author establishes a vanishing line in the \((t-s,s)\)-chart for the \(E_2\)-term of this spectral sequence for \(X=S^0\). More precisely he shows that \(E_2^{s,t}(S^0;\ell)= 0\) if \((t-s,s)\) is above a line of slope \((p^2-p-1)^{-1}\) which intersects the \(s\)-axis at a point close to \(s=2\). He uses this result in conjunction with results on the relationship between the \(\ell\)-Adams filtration and the classical \(H{\mathbb F}_p\)-Adams filtration (that he established in [Trans. Am. Math. Soc. 352, No. 3, 1149-1169 (2000; Zbl 0932.55005)]) to obtain explicit bounds for the \(p\)-exponent of the order of elements in \((\pi_*^S)_{(p)}\) which have trivial \(J\)-Hurewicz image. In case \(*=n\not\equiv 0\;mod\;(2p-2)\) this bound is given by \(\min(4,p)+(2p-1)n/((2p-2)(p^2-p-1))\). This improves a corresponding result of \textit{A. Liulevicius} [Proc. Am. Math. Soc. 14, 972-976 (1963; Zbl 0148.17102)]. For the proof of the vanishing result the author considers the subcomplex \(D\) of \(E= E_1^{*,*}(S^0;\ell)\) consisting of the elements with positive \(H{\mathbb F}_p\)-Adams filtration and uses the homology sequence of the short exact sequence \(D \to E \to E/D\) to reduce the problem to showing that the \((t-s,s)\)-charts for \(H_*(D)\) and \(H_*(E/D)\) respectively have corresponding vanishing lines. To see the latter for the complex \(D\) the author shows that \(D\) is isomorphic to a complex \(C\) which can be described as ``the non-Eilenberg-MacLane part'' of \(E_1(X_1;\ell)\), where \(X_1\) is a model for the first \(H{\mathbb F}_p\)-Adams cover of \(S^0\). The actual vanishing result for \(H_*(D)\) then follows from a calculation of \(H_*(C)\) that the author has given in the above cited paper. To obtain the corresponding result for \(H_*(E/D)\) the author first shows that \(H_*(E/D)\) can be identified with the relative Ext-groups \(\text{Ext}^{*,*}_{A,E}(D)\), where \(A\) is the dual Steenrod algebra and \(E\subset A\) is the exterior algebra generated by \(c(\tau_0)\) and \(c(\tau_1)\), and then he proves a corresponding vanishing result for \(\text{Ext}^{*,*}_{A,E}(M)\) for arbitrary connective \(A\)-comodules \(M\).