\(I_n\)-local Johnson-Wilson spectra and their Hopf algebroids
The Johnson-Wilson homology theories \(E(n)_* (\cdot)\) are a sequence of complex oriented generalized homology theories defined from Brown-Peterson homology using the Landweber exact functor theorem. As with any generalized homology theory the Hopf algebroids \(E(n)_* E(n)\) of stable cooperations for these theories are of great interest because of their central role in Adams spectral sequence calculations. The coefficient rings \(E(n)_*\) of these theories contain maximal ideals \(I_n=(p,v_1,\dots, v_{n-1})\). The main results of this paper are a proof that \(E(n)\), its localization with respect to \(I_n\) and its completion with respect to this ideal are all Bousfield equivalent and a proof that the Hopf algebroid of cooperations in the local case is free as a module over its coefficient ring. The proof of the first result involves a Landweber exact functor argument while the second is proved by means of a direct construction of a basis for each submodule in an increasing filtration starting from a known basis for \(E(n)_*BP\). Unfortunately, it seems to the reviewer that there is a mistake in the proof of Proposition 2.2 of the paper (specifically, the assertion in the 6-th line of the proof that the number of \(\kappa\) elements and the number of \(t\) monomials in \(E(n)_*BP^{[m]}\) are the same fails if \(m = p^n \cdot (p -1)\)). While the argument here can be altered to give a valid proof of this proposition the proof that the localization of \(E(n)_*E(n)\) is free involves results in the proof of proposition 2.2 rather than just its statement and so will require some alterations also.
scientific article; zbMATH DE number 1477086