One-sided Frobenius pairs in extriangulated categories

Cut notions in extriangulated categories related to Auslander-Buchweitz theory and cotorsion theory

“…In [30], H. Nakaoka and Y. Palu introduce the notion of extriangulated categories, which extends well-known results on abelian, exact and triangulated categories. It is also known that AB-theory and cotorsion theory do not scape to generalizations on this context as we can see in the remarkable works done in [27,23,14,35] and [20,13] for such theories, respectively.…”

“…Let us commence with the analogs of left Frobenius pairs and weak Auslander-Buchweitz contexts in extriangulated categories. These notions have been given previously, for example in [4,35,24], however in this paper we work through of a relativization that depends on a class of objects in an extriangulated category C with higher extension groups. Following [35], we first recall the notion of left Frobenius pair in such extriangulated categories.…”

“…These notions have been given previously, for example in [4,35,24], however in this paper we work through of a relativization that depends on a class of objects in an extriangulated category C with higher extension groups. Following [35], we first recall the notion of left Frobenius pair in such extriangulated categories. Now, we extend the notion of cut Frobenius pair, given in [16,Def.…”

“…3.9]. Thus, by proceeding as in [35,Prop. 4.18] and by Lemma 2.4-(4), we obtain that ω ∧ ∩ S = (ω ∩ S) ∧ is closed under extensions, direct summands and cones.…”

“…[35, Def. 4.7] Let C be an extriangulated category with higher extension groups and X , ω ⊆ C. We say that (X , ω) is a left Frobenius pair in C if the following conditions are satisfied.…”