A curious example of triangulated-equivalent model categories which are not Quillen equivalent

Abstract

The paper gives a new proof that the model categories of stable modules for the rings Z=p 2 and Z=pOE =.2 / are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent but whose associated model categories of modules are not Quillen equivalent. As a bonus, we also obtain derived equivalent dgas with non-isomorphic K -theories.18E30, 18F25, 55U35

A tour about existence and uniqueness of dg enhancements and lifts

“…We denote by T p the category Mod(F p ) endowed with the triangulated structure defined by [1] = id (and, necessarily, distinguished triangles given by triangles inducing long exact sequences). We recall the following important result (see [56], [21]).…”

“…On the other hand, the fact that the two model categories are not Quillen equivalent is a deep result, whose original proof in [56] uses rather involved K-theoretical computations. Also the simpler proof in [21] is far from trivial and uses homotopy endomorphism ring spectra.…”

Rigidity and exotic models for $$v_1$$-local G-equivariant stable homotopy theory

“…In this case the same happens except we do not know whether the equivalence Ho(C) ∼ Ho(L 1 Sp) is triangulated. For further examples of exotic models see [12,[34][35][36]41].…”

Rigidity in equivariant stable homotopy theory

“…It then follows from Dugger and Shipley [11,Corollary 3.10] that the model categories Mod-Z=p 2 and Mod-F p OEt=.t 2 / are not Quillen equivalent. Note that there is also a reinterpretation of this example in terms of differential graded algebras by Dugger and Shipley [12].…”