Left Schur subcategories in recollements
Yingying Zhang, Dajun Song
TL;DR
The paper addresses how left Schur subcategories, which unify torsion-free classes and wide subcategories, behave under recollements of length abelian categories. It proves that, when $i^{!}$ is exact, left-Schur structures on the edge categories $\mathcal{Y}$ and $\mathcal{Z}$ are equivalent to the left-Schur structure on the middle category $\mathcal{X}$ via the subcategory $\mathcal{E}_{\mathcal{X}}=\{X\in \mathcal{X} \mid j^{*}X\in \mathcal{E}_{\mathcal{Z}}, i^{!}X\in \mathcal{E}_{\mathcal{Y}}\}$; this is established by transferring monobricks across the recollement and showing $\mathcal{E}_{\mathcal{X}}=\operatorname{Filt}\mathcal{M}_{\mathcal{X}}$ with $\mathcal{M}_{\mathcal{X}}=i_{*}(\mathcal{M}_{\mathcal{Y}})\sqcup j_{*}(\mathcal{M}_{\mathcal{Z}})$. The results extend to wide and torsion-free subcategories and yield explicit cofinally closed monobricks; applications include a transfer principle for left-Schur, wide, and torsion-free properties in triangular matrix algebras, illustrated by an example.
Abstract
Recently left Schur subcategories in a length abelian category were introduced by Enomoto, which unify torsion-free classes and wide subcategories. In this paper, we show the construction of left Schur subcategories in the recollements of length abelian categories. Moreover, we show the construction restricts to wide subcategories and torsion-free classes. As an application, we give an explicit construction of cofinally closed monobricks in recollements.