Left Frobenius Pairs, Cotorsion Pairs and Weak Auslander-Buchweitz Contexts in Triangulated Categories

Let [Formula: see text] be a triangulated category with a proper class [Formula: see text] of triangles. We introduce the notions of left Frobenius pairs, left ([Formula: see text]-)cotorsion pairs and left (weak) Auslander-Buchweitz contexts with respect to [Formula: see text] in [Formula: see text]. We show how to construct left cotorsion pais from left [Formula: see text]-cotorsion pairs, and establish a one-to-one correspondence between left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. Some applications are given in the Gorenstein homological theory of triangulated categories.