Reduction of wide subcategories and recollements

TL;DR

This paper establishes a reduction technique for wide subcategories in abelian categories with recollements, linking subcategories across categories and inducing new recollements, thus generalizing several known reduction methods.

Contribution

It introduces a novel reduction result for wide subcategories in abelian categories with recollements, connecting subcategories across categories and generating new recollements.

Findings

Bijective correspondence between wide subcategories in related categories.

Recollement structure induces new recollements for certain subcategories.

Generalizes Calabi-Yau, silting, and τ-tilting reductions.

Abstract

In this paper, we prove a reduction result on wide subcategories of abelian categories which is similar to Calabi-Yau reduction, silting reduction and $\tau$-tilting reduction. More precisely, if an abelian category $\mathcal{A}$ admits a recollement relative to abelian categories $\mathcal{A}'$ and $\mathcal{A}"$, diagrammatically expressed by $$\xymatrix@!C=2pc{ \mathcal{A'} \ar@{>->}[rr]|{i_{*}} && \mathcal{A} \ar@<-4.0mm>@{->>}[ll]_{i^{*}} \ar@{->>}[rr]|{j^{*}} \ar@{->>}@<4.0mm>[ll]^{i^{!}}&& \mathcal{A''} \ar@{>->}@<-4.0mm>[ll]_{j_{!}} \ar@{>->}@<4.0mm>[ll]^{j_{*}} },$$ then the assignment $\cc\mapsto j^*(\cc)$ defines a bijection between wide subcategories in $\mathcal{A}$ containing $i_{*}(\mathcal{A}')$ and wide subcategories in $\mathcal{A}"$. Moreover, a wide subcategory $\mathcal{C}$ of $\mathcal{A}$ containing $i_{*}(\mathcal{A}')$ admits a new recollement relative to $\mathcal{A}'$ and $j^{*}(\mathcal{C})$ which is induced from the original recollement.