Applications of cotorsion triples

We study homotopy categories of model categories arising from a cotorsion triple, and the equivalences between corresponding stable categories. We characterize homological dimensions with respect to a cotorsion triple. Then, we lift cotorsion triple to complexes, and get the equivalence of homotopy categories of complexes via Quillen equivalence of model categories. Finally, we specify to Gorenstein cotorsion triple $($\mathcal{G}\mathcal{P}$, $\mathcal{W}$, $\mathcal{G}\mathcal{I}$)$. By Quillen equivalence, it is shown that for a left-Gorenstein ring, there is an equivalence $$\mathrm{K}$($\mathcal{G}\mathcal{P}$)\simeq $\mathrm{K}$($\mathcal{G}\mathcal{I}$)$,whichrestrictstoanequivalence$$\mathrm{K}$($\mathcal{P}$)\simeq $\mathrm{K}$($\mathcal{I}$)$ (compare to [J. Algebra, 2010, 324:2718-2731]); a new proof for Bennis and Mahdou’s equality of Gorenstein global dimension is also given.