On relations between 1-lines of Adams-Novikov spectral sequences modulo invariant prime ideals (Q1779510)

For a given prime \(p\) let \(K(n)\) denote the \(n\)-th Morava-\(K\)-theory spectrum, and let \(L_{K(n)}\) and \(L_n\) denote the localization functors associated to \(K(n)\) and \(K(0) \vee \dots \vee K(n)\), respectively. A weak form of Hopkins' chromatic splitting conjecture says that for all type \(n\) finite spectra \(X\) the natural map \(L_n X \to L_nL_{K(n+1)}X\) is a split monomorphism. Correspondingly one would obtain a split monomorphism between the \(E^2\)-terms of the \(BP\)-Adams-Novikov-spectral sequences associated to \(L_n X\) and \(L_nL_{K(n+1)}X\), respectively. Since \(X\) is of type \(n\) one can use change of ring isomorphisms to identify these \(E^2\)-terms with the continuous cohomology groups \(H_c^{**}(G_n; E_{n,*}(X))\) and \(H_c^{**}(G_{n+1};E_{n+1,*}(X))[v_n^{-1}]\). Here \(E_{n,*}\) stands for a Morava \(E\)-theory whose coefficients are given by \(W(F) [[u_1,\dots,u_{n-1}]][u^\pm]\) with \(W(F)\) the ring of Witt vectors for a finite field \(F\) which contains the fields \(F_{p^n}\) and \(F_{p^{n+1}}\), and \(G_n\) is the associated Galois extension of the Morava stabilizer group \(S_n\). In case \(X\) is a model for a Smith-Toda complex \(V(n-1)\) the above cohomology groups can be identified with the continuous cohomology groups \(H_c^{**}(G_n; F[w^{\pm1}])\) and \(H_c^{**}(G_{n+1}; F((u_n))[u^{\pm1}])\), where \(w\) and \(u\) are roots of degree \(-2\) of \(v_n \) and \(v_{n+1}\), respectively. In [\textit{T. Torii}, Am. J. Math. 125, 1037--1077 (2003; Zbl 1046.55003)], the author purely algebraically constructed a corresponding homomorphism \(\Theta: H_c^{1}(G_n; F[w^{\pm1}]) \to H_c^{1}(G_{n+1}; F((u_n))[u^{\pm1}])\) which in the cases where \(V(n-1)\) exists models the corresponding homomorphism induced by the map \(L_n V(n-1) \to L_nL_{K(n+1)}V(n-1)\). Based on further results in [ibid.] the author shows in the paper under review that \(\Theta\) is indeed injective.