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Monoidality of Franke’s exotic model. (English) Zbl 1246.55009
This paper concerns $$\mathrm{Ho}(L_1\mathcal{S})$$, the $$K_{(p)}$$-local stable homotopy category for $$p$$ odd. Here $$L_1\mathcal{S}$$ denotes Bousfield localization with respect to the homology theory $$E(1)$$. J. Franke [“Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence”, preprint, http://www.mathematik.uni-osnabrueck.de/K-theory/0139], constructed an algebraic model category whose homotopy category is equivalent to $$\mathrm{Ho}(L_1\mathcal{S})$$. Franke’s construction, however, does not induce the monoidal structure on the homotopy category. The authors construct a new model structure on Franke’s model category with compatible monoidal product which is Quillen equivalent to the original. They prove the induced product is compatible with the smash product on $$\mathrm{Ho}(L_1\mathcal{S})$$ expanding on results of N. Ganter [Cah. Topol. Géom. Différ. Catég. 48, No. 1, 3–54 (2007; Zbl 1126.18007)]. They also prove the Picard group of the homotopy category of their model is isomorphic to the integers.

##### MSC:
 55P42 Stable homotopy theory, spectra 55P60 Localization and completion in homotopy theory 55U35 Abstract and axiomatic homotopy theory in algebraic topology
##### Keywords:
stable homotopy theory; localizations; model categories
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##### References:
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