A version of Landweber's filtration theorem for \(v_ n\)-periodic Hopf algebroids (Q1911588)

Let \(M_*\) be a \(BP_*(BP)\)-comodule which is finitely presented as a module over \(BP_*\). According to Landweber's filtration theorem \(M_*\) admits a filtration so that each successive subquotient has the form \(BP_*/(p,v_1, \dots, v_{n-1})\) for some \(n\). The proof uses the fact that the grading on \(M_*\) is bounded below. The author asks if there are counterexamples to the filtration theorem for comodules over periodic Hopf algebroids. First he proves a filtration theorem for finitely generated discrete topologized comodules over Hopf algebroids which are associated to Noetherian completions of the spectra \(E(n)\). Finally he exhibits comodules without filtration of Landweber type.