Recollements of Triangulated Categories and the Singularity Category of a Triangular Matrix Category

Introduced by Buchweitz, the singularity category of an algebra $A$ measures its homological singularity, vanishing if and only if $A$ has finite global dimension. This notion extends naturally to the context of $k$-categories. In this paper we study the singularity category of a triangular matrix category $\mathrm{\Lambda }:=\left[\begin{array}{cc}\mathcal{T}& 0\\ M& \mathcal{U}\end{array}\right]$. By utilizing the framework of recollements, we provide a characterization of this category, proving that when certain homological conditions are satisfied, there exists an equivalence of singularity categories $D_{sg}(\mathrm{M}\mathrm{o}\mathrm{d}(\mathrm{\Lambda }))\simeq D_{sg}(\mathrm{M}\mathrm{o}\mathrm{d}(\mathcal{U}))$. This result generalizes the one obtained by Pin Liu and Ming Lu in [16].