Triangular Matrix Categories I: Dualizing Varieties and generalized one-point extension

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 additive categories $\mathcal{U}$ and $\mathcal{T}$ and $M\in \mathsf{Mod}(\mathcal{U}\otimes \mathcal{T}^{op})$ we construct the triangular matrix category $\mathbf{\Lambda}:=\left[\begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$. First, we prove that there is an equivalence $\Big( \mathsf{Mod}(\mathcal{T}), \mathbb{G}\mathsf{Mod}(\mathcal{U})\Big) \simeq \mathrm{Mod}(\mathbf{\Lambda})$. 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 $\mathbf{\Lambda}:=\left[\begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$ is a dualizing variety (see theorem 6.10). In particular, $\mathrm{mod}(\mathbf{\Lambda})$ 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.