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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.

##### MSC:
 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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