Thick subcategories and silting subcategories in recollement

TL;DR

This paper explores the relationships between thick and silting subcategories within recollement structures of extriangulated categories, establishing bijections and methods to construct new recollements and silting subcategories.

Contribution

It introduces a bijection between thick subcategories in related categories and shows how silting subcategories can be glued across recollements, providing new insights into their structure.

Findings

Bijection between thick subcategories in $\\mathcal{C}$ and those in $\mathcal{B}$ containing $i_{\ast}\mathcal{A}$.

Construction of new recollements from thick subcategories containing $i_{\ast}\mathcal{A}$.

Silting subcategories can be glued across recollements, with conditions for the converse.

Abstract

Let $(\mathcal{A}, \mathcal{B}, \mathcal{C}, i^{*}, i_{\ast}, i^{!},j_!, j^\ast, j_\ast)$ 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.