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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.

##### MSC:
 55P42 Stable homotopy theory, spectra
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