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

Abstract

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 $Λ:=\left[ \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$. 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 $D_{sg}(\mathrm{Mod(Λ)})\simeq D_{sg}(\mathrm{Mod}(\mathcal{U}))$. This result generalizes the one obtained by Pin Liu and Ming Lu in [16].