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

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 $\mathsf{M}\mathsf{o}\mathsf{d}$($\mathcal{U}$\otimes $\mathcal{T}$^{op})$ we construct the triangular matrix category $$\mathbf{\Lambda }$:=\left[\begin{smallmatrix} $\mathcal{T}$ & 0 \ M & $\mathcal{U}$ \end{smallmatrix}\right]$. First, we prove that there is an equivalence $\Big( $\mathsf{M}\mathsf{o}\mathsf{d}$($\mathcal{T}$), $\mathbb{G}$$\mathsf{M}\mathsf{o}\mathsf{d}$($\mathcal{U}$)\Big) \simeq $\mathrm{M}\mathrm{o}\mathrm{d}$($\mathbf{\Lambda }$)$\mathrm{.}Oneofourmainresultsisthatif$$\mathcal{U}$$and$$\mathcal{T}$$aredualizing$K$-varietiesand$M\in $\mathsf{M}\mathsf{o}\mathsf{d}$($\mathcal{U}$\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, $\mathrm{m}\mathrm{o}\mathrm{d}(\mathbf{\Lambda })$ 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.