ICE-closed subcategories and epibricks over recollements

TL;DR

This paper studies how ICE-closed subcategories and epibricks in abelian categories relate within recollements, establishing bijections and new recollement structures that extend these subcategories.

Contribution

It proves that ICE-closed subcategories and epibricks can be extended across recollements and introduces a new recollement structure for certain ICE-closed subcategories.

Findings

Extension of ICE-closed subcategories across recollements.

Bijection between ICE-closed subcategories in related categories.

Construction of a new recollement from ICE-closed subcategories.

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 $j_!{j^\ast(\mathcal{C})}\subset\mathcal{C}$.