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Assume that the class of all Gorenstein (ℒ,풜)-flat modules is closed under extensions. We define a notion of Gorenstein (ℒ,풜)-flat dimension for complexes and consider equivalent characterizations of the finiteness of Gorenstein (ℒ,풜)-flat dimension of complexes. Furthermore, in addition, assume that R is a right coherent ring. We construct the relative singularity category with respect to Gorenstein (ℒ,풜)-flat modules, as the triangulated quotient of the triangulated subcategory of Db(R) consisting of all complexes with both finite Gorenstein (ℒ,풜)-flat dimension and cotorsion dimension by the bounded homotopy category of flat-cotorsion modules, and show that the relative singularity category is triangulated equivalent to the stable category of the Frobenius category consisting of all Gorenstein (ℒ,풜)-flat and cotorsion modules. [ABSTRACT FROM AUTHOR]

In this paper, we construct some model structures corresponding Gorenstein (ℒ ,) -modules and relative Gorenstein flat modules associated to duality pairs, Frobenius pairs and cotorsion pairs. By investigating homological properties of Gorenstein (ℒ ,) -modules and some known complete hereditary cotorsion pairs, we describe several types of complexes and obtain some characterizations of Iwanaga–Gorenstein rings. Based on some facts given in this paper, we find new duality pairs and show that ℐ is covering as well as enveloping and ℱ is preenveloping under certain conditions, where ℐ denotes the class of Gorenstein (ℒ ,) -injective modules and ℱ denotes the class of Gorenstein (ℒ ,) -flat modules. We give some recollements via projective cotorsion pair ( , ⊥) cogenerated by a set, where denotes the class of Gorenstein (ℒ ,) -projective modules. Also, many recollements are immediately displayed through setting specific complete duality pairs. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the homotopy categories K (𝒳 ∩ 𝒴) and K (𝒳 ∩ 𝒴 ̃) with respect to a complete and hereditary cotorsion pair (𝒳 , 𝒴) in a bicomplete abelian category. We prove that K (𝒳 ∩ 𝒴 ̃) is compactly generated provided that K (𝒳 ∩ 𝒴) is compactly generated. We introduce and characterize Gorenstein 𝒳 ̃ -complexes with respect to (𝒳 ̃ , d g 𝒴 ̃) and show that a complex is a Gorenstein 𝒳 ̃ -complex if and only if its every term is a Gorenstein 𝒳 -object. We also show that the inclusion functors K (𝒳 ∩ 𝒴) ↪ K (𝒢 (𝒳)) and K (𝒳 ∩ 𝒴 ̃) ↪ K (𝒢 (𝒳 ̃)) have right adjoints respectively when K (𝒳 ∩ 𝒴) is compactly generated. [ABSTRACT FROM AUTHOR]

In this paper, we give some recollements of additive categories associated to concentric twin cotorsion pairs in a triangulated category. Some applications are given and Krause's recollement is obtained with a new method. [ABSTRACT FROM AUTHOR]