ICE-closed subcategories and epibricks over recollements

Abstract

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