[Paper Review] Recollement of additive quotient categories

Minxiong Wang, Zengqiang Lin|arXiv (Cornell University)|Feb 2, 2015

Algebraic structures and combinatorial models11 references3 citations

TL;DR

This paper introduces a recollement structure for additive categories and proves that such a recollement induces a recollement in their quotient categories under suitable conditions. The key contribution is establishing a recollement of quotient triangulated categories from mutation pairs in triangulated categories, unifying abelian and triangulated recollement frameworks.

ABSTRACT

In this note, we define a recollement of additive categories, and prove that such a recollement can induce a recollement of their quotient categories. As an application, we get a recollement of quotient triangulated categories induced by mutation pairs.

Motivation & Objective

• To define a recollement structure for additive categories, generalizing existing recollements in abelian and triangulated categories.
• To establish conditions under which a recollement of additive categories induces a recollement in their quotient categories.
• To apply the general framework to triangulated categories, particularly via mutation pairs, to construct recollements of quotient triangulated categories.
• To unify the treatment of recollements across abelian and triangulated settings using quotient constructions.

Proposed method

• Define a recollement of additive categories via six functors satisfying adjoint and full embedding conditions.
• Use quotient category construction: for an additive category 𝒜 and subcategory 𝒳, define 𝒜/𝒳 as the category with morphisms modulo those factoring through 𝒳.
• Prove that functors between additive categories preserving the subcategory structure induce functors between quotient categories.
• Show that the induced functors in the quotient categories preserve adjunctions and full embeddings, thus forming a recollement.
• Apply the framework to triangulated categories by assuming (𝒞,𝒞) is a 𝒟-mutation pair with 𝒟 ⊆ Ker j*.
• Leverage known results on quotient triangulated categories (e.g., Iyama-Yoshino) to ensure the quotient category 𝒞/𝒟 is triangulated.

Experimental results

Research questions

• RQ1 Can a recollement of additive categories induce a recollement in their quotient categories under appropriate conditions?
• RQ2 How can recollement structures be consistently defined and preserved under quotient constructions in additive categories?
• RQ3 Does a recollement of triangulated categories induce a recollement in the quotient categories when the subcategory is a mutation pair?
• RQ4 Can the framework unify recollements in abelian and triangulated categories via quotient constructions?
• RQ5 What conditions ensure that the quotient of a triangulated category by a mutation pair remains triangulated and admits a recollement structure?

Key findings

• A recollement of additive categories induces a recollement in the quotient categories provided the subcategories are preserved under the involved functors.
• The quotient functors between additive categories lift to functors between quotient categories, preserving adjunctions and full embeddings.
• When (𝒞,𝒞) is a 𝒟-mutation pair and 𝒟 ⊆ Ker j*, the quotient category 𝒞/𝒟 inherits a triangulated structure.
• The induced functors in the quotient category form a recollement of triangulated categories, with exact functors between them.
• The result extends to functorially finite subcategories closed under the Auslander-Reiten translation, ensuring the quotient remains triangulated.