Reduction of wide subcategories and recollements

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.