Rigidity and Exotic Models for the <i>K</i>-local Stable Homotopy Category

Abstract

When two stable model categories C and D are Quillen equivalent, their homotopy categories Ho(C) and Ho(D) are equivalent as triangulated categories. But is the converse also true? <br /> For the stable homotopy category Ho(Sp), i.e., the homotopy category of spectra, there is the following result by Stefan Schwede:<br /> Rigidity Theorem (Schwede `05) Let C be a stable model category, and f: Ho(Sp) -> Ho(C) an equivalence of triangulated categories. Then the underlying model categories Sp and C are Quillen equivalent, i.e., Ho(Sp) is ''rigid''.<br /> Next, we ask the question if this is also true for Bousfield localisations of the stable homotopy category at a generalised cohomology theory. This thesis treats the case of Ho(L_K Sp), i.e. the stable homotopy category localised at 2-local complex K-theory K_(2) and comes to the conclusion that Ho(L_K Sp) is rigid:<br /> K_(2)-local Rigidity Theorem (Roitzheim) Let C be a stable model category, and f: Ho(L_K Sp) -> Ho(C) an equivalence of triangulated categories. Then the underlying model categories L_K Sp and C are Quillen equivalent, i.e., Ho(L_1 Sp) is ''rigid''.<br /> However, for odd primes p, the K_(p)-local stable homotopy category is not rigid, and we discuss a counterexample given by Jens Franke.