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\(I_n\)-local Johnson-Wilson spectra and their Hopf algebroids. (English) Zbl 0946.55002
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.

55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
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