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The paper gives a new proof that the model categories of stable modules forthe rings Z/(p^2) and (Z/p)[\epsilon]/(\epsilon^2) are not Quillen equivalent.The proof uses homotopy endomorphism ring spectra. Our considerations lead toan example of two differential graded algebras which are derived equivalent butwhose associated model categories of modules are not Quillen equivalent. As abonus, we also obtain derived equivalent dgas with non-isomorphic K-theories.

A curious example of two model categories and some associated differential graded algebras