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By Daniel (-or) Shipley and Mr Mr (e


A curious example of triangulated-equivalent model categories which are not Quillen equivalent. (English summary) Algebr. Geom. Topol. 9 (2009), no. 1, 135–166. Recall that two rings, or more generally, two DGAs (differential graded algebras) are derived equivalent if their derived categories are equivalent as triangulated categories. They are Quillen equivalent if their categories of differential graded modules are weakly equivalent as model categories. Quillen equivalence implies derived equivalence, and the converse is true for rings [D. Dugger and B. E. Shipley, Duke Math. J. 124 (2004), no. 3, 587–617; MR2085176 (2005e:19005)]. One may say that rings have no higher homotopy invariants. In this paper, the authors show that DGAs can have higher homotopy invariants by finding two derived equivalent DGAs that are not Quillen equivalent. These two DGAs also have different algebraic K-theories even though they are derived equivalent, which again cannot happen with rings. The counterexample in question is built from an example of M. Schlichting [Invent. Math. 150 (2002), no. 1, 111–116; MR1930883 (2003h:18015)] of two very simple model categories with equivalent homotopy categories but different algebraic K-theories, the stable module categories of the quasi-Frobenius rings Z/p 2 and Z/p[e]/(e 2). One of the main points of this paper is that these model categories are each Quillen equivalent to differential graded modules over a DGA, namely, the endomorphism DGA of a Tate resolution of Z/p

Year: 2013
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