Triangular Matrix Categories I: Dualizing Varieties and Generalized One-point Extensions

Abstract

Following Mitchell’s philosophy, in this paper we define the analogous of the triangular matrix algebra to the context of rings with several objects. Given two preadditive categories $\mathcal {U}$ and $\mathcal {T}$ and $M\in \mathsf {Mod}(\mathcal {U}\otimes \mathcal {T}^{op})$ we construct the triangular matrix category $\mathbf {{\varLambda }}:=\left [\begin {array}{ll} \mathcal {T} & 0 \\ M & \mathcal {U} \end {array}\right ]$. First, we prove that Mod(Λ) is equivalent to a comma category $(\mathsf {Mod}(\mathcal {T}), \mathbb {G}\mathsf {Mod}(\mathcal {U}))$ which is induced by a functor $\mathbb {G}: \mathsf {Mod}(\mathcal {U})\rightarrow \text {Mod}(\mathcal {T})$. One of our main results is that if $\mathcal {U}$ and $\mathcal {T}$ are dualizing K-varieties and $M\in \mathsf {Mod}(\mathcal {U}\otimes \mathcal {T}^{op})$ satisfies certain conditions then $\mathbf {{\varLambda }}:=\left [\begin {array}{ll} \mathcal {T} & 0 \\ M & \mathcal {U} \end {array}\right ]$ is a dualizing variety (see Theorem 6.10). In particular, mod(Λ) has Auslander-Reiten sequences. Finally, we apply the theory developed in this paper to quivers and give a generalization of the so called one-point extension algebra.