Left Frobenius pairs, cotorsion pairs and weak Auslander-Buchweitz contexts in triangulated categories (Q6546518)

Let \(\mathcal T\) be a triangulated category with a proper class \(\xi\) of triangles. The authors introduce the concepts of left Frobenius pairs, left \(n\)-cotorsion pairs, and left (weak) Auslander-Buchweitz contexts with respect to \(\xi\) in \(\mathcal T\). They demonstrate how to construct left cotorsion pairs from left \(n\)-cotorsion pairs and establish a one-to-one correspondence between left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. Additionally, the authors provide applications of these results in the Gorenstein homological theory of triangulated categories.\N\NOverall, their work extends the theoretical foundations of homological algebra in triangulated categories, providing new methods and results that can be applied to a wide range of problems in mathematics.