2-recollements of singualrity categories and Gorenstein defect categories over triangular matrix algebras

TL;DR

This paper investigates the conditions under which 2-recollement structures of singularity and Gorenstein defect categories over triangular matrix algebras exist, linking them to the properties of the corner algebras.

Contribution

It provides necessary and sufficient conditions for 2-recollements of these categories over triangular matrix algebras, extending and unifying previous results.

Findings

Established criteria for 2-recollements over triangular matrix algebras.

Linked recollement structures to properties of corner algebras.

Unified previous partial results in the literature.

Abstract

Let $T=(A,M,0,B)$ be a triangular matrix algebra with its corner algebras $A$ and $B$ Artinian and $_AM_B$ an $A$-$B$-bimodule. The 2-recollement structures for singularity categories and Gorenstein defect categories over $T$ are studied. Under mild assumptions, we provide necessary and sufficient conditions for the existences of 2-recollements of singularity categories and Gorenstein defect categories over $T$ relative to those of $A$ and $B$. Parts of our results strengthen and unify the corresponding work in [27,28,34].