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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
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