Applications of Cotorsion Pairs on Triangulated Categories

Giving a cotorsion pair in an abelian category $${\mathscr {C}}$$ , we have a sequence of exact functors between triangulated categories with respect to the pair, and construct right (left) adjoints of the exact functors such that the sequence is a (co)localization sequence. Further, for some especial cotorsion pairs, we gain a recollement and triangle-equivalences between corresponding triangulated categories. In particular, let ( $${\mathcal {A}}, {\mathcal {Z}}, {\mathcal {B}}$$ ) and ( $${\mathcal {A}}_{1}, {\mathcal {Z}}_{1}, {\mathcal {B}}_{1}$$ ) be two complete and hereditary cotorsion triples in $${\mathscr {C}}$$ with $${\mathcal {A}}_{1}\subseteq {\mathcal {A}}$$ . We obtain a triangle-equivalence $$K({\mathcal {A}})\simeq K({\mathcal {B}})$$ , which restricts to an equivalence $$K({\mathcal {A}}_{1})\simeq K({\mathcal {B}}_{1})$$ .