Let $\mathscr{C}$ be a extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. We introduce the notions of left Frobenius pairs, left ($n$-)cotorsion pairs and left (weak) Auslander-Buchweitz contexts with respect to $\xi$ in $\mathscr{C}$. We show how to construct left cotorsion pais from left $n$-cotorsion pairs, and establish a one-one correspondence between left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. This generalizes Huang-Ma-Zhao's results in triangulated categories and partially generalizes Becerril-Mendoza-P\'{e}rez-Santiago's results in abelian categories.

One-sided Frobenius pairs in extriangulated categories

Recently, Wang, Wei and Zhang introduced the notion of recollements of extriangulated categories. In this paper, let $$({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$$ be a recollement of extriangulated categories. We provide some methods to construct resolving subcategories involved in a recollement and study how they are related. As applications of the Auslander–Reiten correspondence, we get the gluing of cotilting modules in a recollement of module categories for artin algebras. We also give some bounds of resolution dimensions of the categories involved in $$({\mathcal {A}},{\mathcal {B}},{\mathcal {C}})$$ with respect to resolving subcategories, which generalize some known results. 

Resolving subcategories and dimensions in recollements of extriangulated categories

In this paper, let $(\mathcal{A},\mathcal{B},\mathcal{C})$ be a recollement of extriangulated categories. We introduce the notion of resolving subcategories of extriangulated categories, and give some upper bounds of resolution dimensions of the categories involved in $(\mathcal{A},\mathcal{B},\mathcal{C})$ with respect to resolving subcategories.

Resolving subcategories in Extriangulated Categories and Recollements.

Let $\mathscr{C}$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. We introduce the notions of left Frobenius pairs, left ($n$-)cotorsion pairs and left (weak) Auslander-Buchweitz contexts with respect to $\xi$ in $\mathscr{C}$. We show how to construct left cotorsion pais from left $n$-cotorsion pairs, and establish a one-to-one correspondence between left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. We also study the relation between a certain class of cotorsion pairs and that of $n$-cotorsion pairs. These work generalize Ma-Zhao-Huang's results in triangulated categories and partially generalize Becerril-Mendoza-P\'{e}rez-Santiago's results in abelian categories. 

/pdf/one-sided-frobenius-pairs-in-extriangulated-categories-22qk7sqo.pdf

/pdf/resolving-resolution-dimensions-in-triangulated-categories-3u1bdbcbk6.pdf

Resolving resolution dimensions in triangulated categories

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Resolving resolution dimensions in triangulated categories