[Paper Review] Recollements of extriangulated categories

Li Wang, Jiaqun Wei+1 more|arXiv (Cornell University)|Dec 6, 2020

Algebraic structures and combinatorial models8 references4 citations

TL;DR

This paper introduces recollements of extriangulated categories as a unifying framework generalizing recollements in abelian and triangulated categories. It establishes conditions under which cotorsion pairs in the outer categories $π$ and $σ$ induce a cotorsion pair in the middle category $β$, recovering known results in triangulated categories and revealing new phenomena in abelian categories.

ABSTRACT

We give a simultaneous generalization of recollements of abelian categories and triangulated categories, which we call recollements of extriangulated categories. For a recollement $(\mathcal{A}$, $\mathcal{B}$, $\mathcal{C})$ of extriangulated categories, we show that cotorsion pairs in $\mathcal{A}$ and $\mathcal{C}$ induce cotorsion pairs in $\mathcal{B}$ under certain conditions. As an application, our main result recovers a result given by Chen for recollements of triangulated categories, and it also shows a new phenomena when it is applied to abelian categories.

Motivation & Objective

• To unify the concepts of recollements in abelian and triangulated categories by introducing recollements in the broader context of extriangulated categories.
• To investigate the relationship between cotorsion pairs in the outer categories of a recollement and their potential to induce a cotorsion pair in the middle category.
• To generalize Chen's result on recollements of triangulated categories and reveal new behaviors when applied to abelian categories.
• To establish sufficient conditions under which a glued pair of cotorsion pairs in $π$ and $σ$ forms a cotorsion pair in $β$.
• To explore the converse: when a cotorsion pair in $β$ arises from cotorsion pairs in $π$ and $σ$ under specific conditions.

Proposed method

• Introduce and formalize the notion of a recollement in extriangulated categories using compatible morphisms and the WIC condition from Nakaoka and Palu.
• Define left and right exact functors, left and right exact sequences, and compatible morphisms within the framework of extriangulated categories.
• Establish the gluing of cotorsion pairs via the functors $i^*$, $i^!$, $j_*$, $j^*$, and $j^!$ associated with the recollement structure.
• Use the vanishing of $τ$-extensions and $σ$-extensions to verify the existence of cotorsion pairs in the middle category $β$.
• Apply the main theorem to recover Chen's result on recollements of triangulated categories as a special case.
• Provide explicit examples in mod-$A$ and mod-$B$ to illustrate the gluing process and verify the conditions of the main theorem.

Experimental results

Research questions

• RQ1 Under what conditions does a pair of cotorsion pairs in the outer categories $π$ and $σ$ of a recollement of extriangulated categories induce a cotorsion pair in the middle category $β$?
• RQ2 How does the proposed framework of recollements in extriangulated categories generalize existing results in abelian and triangulated categories?
• RQ3 Can the converse hold: when does a cotorsion pair in $β$ arise from cotorsion pairs in $π$ and $σ$?
• RQ4 What new phenomena emerge when applying the main result to abelian categories, as opposed to triangulated categories?
• RQ5 How do the functors $i^*$, $i^!$, $j^*$, and $j^!$ interact with cotorsion pairs in the context of extriangulated recollements?

Key findings

• The paper establishes that if $(π_1, π_2)$ and $(σ_1, σ_2)$ are cotorsion pairs in $π$ and $σ$, respectively, and certain conditions on the functors $i^*$, $i^!$, $j^*$, and $j^!$ are satisfied, then the glued pair $(β_1, β_2)$ forms a cotorsion pair in $β$.
• The main result recovers Chen's theorem on recollements of triangulated categories as a special case, confirming consistency with prior work.
• When applied to abelian categories, the framework reveals new phenomena not present in the triangulated case, indicating richer structure in the abelian setting.
• The example in mod-$B$ shows that not all glued pairs are cotorsion pairs, as $τ$-extensions may not vanish, but specific choices (e.g., $τ = π(ρ)$, $σ = ρ(ρ)$) do yield valid cotorsion pairs.
• The paper demonstrates that $i_*i^*τ o τ$ and $j_*j^*σ o σ$ being in the respective subcategories is sufficient for the induced pairs $(i^*τ, i^!σ)$ and $(j^*τ, j^*σ)$ to be cotorsion pairs in $π$ and $σ$.
• In Example 4.7, it is shown that while some glued pairs fail to be cotorsion pairs due to nonvanishing $τ$-extensions, others—such as $(π(ρ), ρ(ρ))$—do satisfy the conditions and are valid cotorsion pairs in mod-$B$.