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Invertible spectra in the \(E(n)\)-local stable homotopy category. (English) Zbl 0947.55013
Let \(E(n)\) be the Johnson-Wilson spectrum at a prime \(p\) and \({\mathcal L}\) denote the category of \(E(n)\)-local spectra. \(\text{Pic} ({\mathcal L})\) denotes the Picard group which consists of invertible spectra in \({\mathcal L}\). The main theorem says that there is an isomorphism \(\text{Pic} ({\mathcal L}) \cong\mathbb{Z}\) if \(2p-2> n^2+n\). The condition on \(p\) and \(n\) is essential, for they showed that \(\text{Pic} ({\mathcal L})\cong \mathbb{Z}\oplus \mathbb{Z}/2\) for \(p=2\) and \(n=1\). In fact, the main theorem is shown by the fact that the isomorphism \(E(n)_*(X)\cong E(n)_*\) as an \(E(n)_*(E(n))\)-comodule implies \(X= S^0\) if all the differentials of the \(E(n)\)-Adams spectral sequence for \(X\) are trivial.
In this paper, the authors also show two proofs of a generalization of the Miller-Ravenel change-of-rings theorem: The map \(\text{Ext}^{**}_{B P_* BP}(BP_*,M) \to\text{Ext}^{*,*}_{E(n)_* E(n)}(E(n)_*\), \(E(n)_* \otimes_{B P_*}M)\) is an isomorphism for a \(BP_*BP\)-comodule \(M\) on which \(v_j\) acts isomorphically for a \(j\leq n\). One of them is based on Theorem B which gives isomorphisms \[ L_{K(j)} BP\cong L_{K(j)} \Bigl(\bigvee_I \Sigma^{r(I)} E(j) \Bigr) \cong L_{K(j)} \Bigl(\bigvee_I \Sigma^{r(I)} L_{K(j)} E(j)\Bigr). \] This was conjectured by the first author and originally proved as a part of the paper of Ando-Morava-Sadofsky. They further show a similar splitting for \(E(n)\) instead of \(BP\). The other proof uses an algebraic result of Hopkins. They also give a proof of the result by using a language of homological algebra. This algebraic proof is interesting for the theory involved in it.

55P42 Stable homotopy theory, spectra
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