Thick subcategories and silting subcategories in recollement

Abstract

Let $(\mathcal{A},\mathcal{B},\mathcal{C},i^{\ast },i_{\ast },i^{!},j_{!},j^{\ast },j_{\ast })$ be a recollement of extriangulated categories. We show that there is a bijection between thick subcategories in $\mathcal{C}$ and thick subcategories in $\mathcal{B}$ containing $i_{\ast }\mathcal{A}$. Futhermore, the thick subcategories $\mathcal{V}$ in $\mathcal{B}$ containing $i_{\ast }\mathcal{A}$ can induce a new recollement relative to $\mathcal{A}$ and $j^{\ast }\mathcal{V}$. We also prove that silting subcategories in $\mathcal{A}$ and $\mathcal{C}$ can be glued to get silting subcategories in $\mathcal{B}$ and the converse holds under certain conditions.

19 pages