Triangulated equivalence between a homotopy category and a triangulated quotient category
\textit{M. Hovey} [Math. Z. 241, No. 3, 553--592 (2002; Zbl 1016.55010)] introduced the notion of abelian model categories as bicomplete abelian categories possessing a compatible model structure. His most wonderful result, which is now known as \textit{Hovey's correspondence}, says that there is a one-to-one correspondence between abelian model structures and complete cotorsion pairs. Given two complete hereditary cotorsion pairs \((\mathcal{Q}, \mathcal{R})\) and \((\mathcal{Q}', \mathcal{R}')\) in a bicomplete abelian category \(\mathcal{G}\) such that \(\mathcal{Q}'\subseteq\mathcal{Q}\) and \(\mathcal{Q}\cap\mathcal{R} = \mathcal{Q}'\cap\mathcal{R}'\), \textit{H. Becker} [Adv. Math. 254, 187--232 (2014; Zbl 1348.16009)] showed that there exists a hereditary abelian model structure \(\mathcal{M}=(\mathcal{Q},\mathcal{W},\mathcal{R}')\) on \(\mathcal{G}\), where \(\mathcal{W}\) is a thick subcategory of \(\mathcal{G}\). The authors prove that the homotopy category \(\text{Ho}(\mathcal{M})\) of \(\mathcal{M}\) is triangulated equivalent to the triangulated quotient category \(D^b(\mathcal{G})_{\widehat{[\mathcal{Q},\mathcal{R}']}}/K^b(\mathcal{Q}\cap\mathcal{R}')\), where \(D^b(\mathcal{G})_{\widehat{[\mathcal{Q},\mathcal{R}]}}\) is the subcategory of \(D^b(\mathcal{G})\) consisting of all homology bounded complexes with both finite \(\mathcal{Q}\) dimension and \(\mathcal{R}'\) dimension and \(K^b(\mathcal{Q}'\cap\mathcal{R})\) is the bounded homotopy category of \(\mathcal{Q}\cap\mathcal{R}\) (core) objects. The authors focus their attention on the category of modules for giving some applications of main results. It is shown that the homotopy category of the Gorenstein flat (resp., Ding projective and Gorenstein \(AC\)-projective) model structure on the category of modules established by \textit{J. Gillespie} [Homology Homotopy Appl. 12, No. 1, 61--73 (2010; Zbl 1231.16005)] can be realized as a certain triangulated quotient category.
scientific article; zbMATH DE number 6894500
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