Triangulated equivalence between a homotopy category and a triangulated quotient category

Given two complete hereditary cotorsion pairs (Q,R) and (Q′,R′) in a bicomplete abelian category G such that Q′⊆Q and Q∩R=Q′∩R′, Becker showed that there exists a hereditary abelian model structure M=(Q,W,R′) on G, where W is a thick subcategory of G. We prove that the homotopy category Ho(M) of M is triangulated equivalent to the triangulated quotient category Db(G)[Q,R′]ˆ/Kb(Q′∩R′), where Db(G)[Q,R′]ˆ is the subcategory of Db(G) consisting of all homology bounded complexes with both finite Q dimension and R′ dimension and Kb(Q′∩R′) is the bounded homotopy category of Q′∩R′ (core) objects. Applications are given in the category of modules. 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 Gillespie and his coauthors can be realized as a certain triangulated quotient category.