Rigidity of the \(K(1)\)-local stable homotopy category (Q2326315)

In this paper, the author proves that the \(K(1)\)-local stable homotopy category \(\mathrm{Ho}(L_{K(1)} \mathrm{Sp}) \), for \(K(1)\) the first Morava \(K\)-theory at the prime \(p=2\), is rigid; i.e., if \(\mathcal{C}\) is another stable model category with homotopy category \(\mathrm{Ho} (\mathcal{C})\) that is equivalent to \(\mathrm{Ho}(L_{K(1)} \mathrm{Sp}) \) as a triangulated category, then \(\mathcal{C}\) is Quillen equivalent to the underlying model category \(L_{K(1)} \mathrm{Sp}\). This extends [\textit{C. Roitzheim}, Geom. Topol. 11, 1855--1886 (2007; Zbl 1142.55007)], which showed that the \(E(1)\)-local stable homotopy category is rigid at the prime \(2\). The author uses the fact that the mod-\(2\) Moore spectrum \(M\) gives a compact generator of the \(K(1)\)-local stable homotopy category, together with the following new characterization of \(K(1)\)-locality: a \(2\)-complete spectrum \(Y\) is \(K(1)\)-local if the mod-\(2\) Adams map \(v_1^4\) induces an isomorphism \([M, Y]_* \stackrel{\cong}{\rightarrow} [\Sigma ^8 M, Y]_*\) of graded morphism groups in the stable homotopy category \(\mathrm{Ho}(\mathrm{Sp})\). By general results, the image \(X\) in \(\mathrm{Ho} (\mathcal{C})\) of the \(K(1)\)-local sphere \(L_{K(1)} \mathbb{S}^0\) induces an adjunction \[ X \wedge^L - : \mathrm{Ho} (\mathrm{Sp}) \rightleftarrows \mathrm{Ho}(\mathcal{C}) : \mathrm{RHom}(X, -). \] The above characterization of \(K(1)\)-locality is used to show that \(X \wedge^L -\) factors across \(\mathrm{Ho}(L_{K(1)} \mathrm{Sp}) \), using properties of the endomorphism ring of \(M\) in \(\mathrm{Ho}(L_{K(1)} \mathrm{Sp}) \), which follow from the \(E(1)\)-local case. The proof that \(X \wedge ^L -\) gives the required Quillen equivalence is similar to that of the \(E(1)\)-local case, using information from \(\pi_* (L_{K(1)} \mathbb{S}^0)\).