ICE-closed subcategories and epibricks over recollements

TL;DR

The paper investigates how ICE-closed subcategories, epibricks and monobricks behave under recollements of abelian categories. It establishes that ICE-closed subcategories from the outer corners extend to the middle and that a bijection via the middle restriction $j^*$ links ICE-closed subcategories in the middle to those on the right-hand side under suitable containment. It also develops a gluing mechanism for epibricks and monobricks using the intermediate extension $j_{!*}$ and, under exactness hypotheses, shows how to form new recollements within ICE-closed subcategories. The results generalize prior gluing/reduction theories to ICE-closed subcategories and bricks, with concrete applications to triangular matrix algebras, one-point extensions, and Morita rings, providing practical tools for constructing such subcategories in composite settings.

Abstract

Let $( \mathcal{A^{'}},\mathcal{A},\mathcal{A^{''}},i^\ast,i_\ast,i_!,j_!,j^\ast,j_\ast)$ be a recollement of abelian categories. We proved that every ICE-closed subcategory (resp. epibrick, monobrick) in $\mathcal{A^{'}}$ or $\mathcal{A^{''}}$ can be extended to an ICE-closed subcategories (resp. epibrick, monobrick) in $\mathcal{A}$, and the assignment $\mathcal{C}\mapsto j^*(\mathcal{C})$ defines a bijection between certain ICE-closed subcategories in $\mathcal{A}$ and those in $\mathcal{A}''$. Moreover, the ICE-closed subcategory $\mathcal{C}$ of $\mathcal{A}$ containing $i_\ast(\mathcal{A^{'}})$ admits a new recollement relative to ICE-closed subcategories $\mathcal{A^{'}}$ and $j^\ast(\mathcal{C})$ which induced from the original recollement when $j_!{j^\ast(\mathcal{C})}\subset\mathcal{C}$.

Theorems & Definitions (42)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Lemma 2.6
• Lemma 3.1
• proof
• Lemma 3.2
• proof