Applications of Cotorsion Pairs on Triangulated Categories

Abstract

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 $K({\mathcal {A}}_{1})\simeq K({\mathcal {B}}_{1})$.