Gorenstein defect categories of triangular matrix algebras

TL;DR

This paper uses recollement techniques to analyze Gorenstein defect categories of triangular matrix algebras, providing categorical insights and descriptions for special cases like simple gluing algebras.

Contribution

It constructs recollements of Gorenstein defect categories for triangular matrix algebras, offering new categorical interpretations and descriptions for specific algebra classes.

Findings

Constructed a left recollement of Gorenstein defect categories under certain conditions.

Provided a categorical interpretation of Gorenstein properties for triangular matrix algebras.

Described singularity and Gorenstein defect categories for simple gluing algebras.

Abstract

We apply the technique of recollement to study the Gorenstein defect categories of triangular matrix algebras. First, we construct a left recollement of Gorenstein defect categories for a triangular matrix algebra under some conditions, using it, we give a categorical interpretation of the Gorenstein properties of the triangular matrix algebra obtained by X-W. Chen, B. L. Xiong and P. Zhang respectively. Second, under some additional conditions, a recollement of Gorenstein defect categories for a triangular matrix algebra is constructed. As an application, for a special kind of triangular matrix algebras, which are called simple gluing algebras, we describe their singularity categories and Gorenstein defect categories.