[Paper Review] Triangular Matrix Categories I: Dualizing Varieties and generalized one-point extension

Alicia León-Galeana, M. Ortiz-Morales+1 more|arXiv (Cornell University)|Mar 10, 2019

Algebraic structures and combinatorial models44 references4 citations

TL;DR

This paper introduces triangular matrix categories for additive categories, generalizing triangular matrix algebras to rings with several objects. It establishes that if two additive categories are dualizing varieties and a bimodule satisfies certain conditions, the resulting triangular matrix category is also a dualizing variety, ensuring the existence of Auslander–Reiten sequences. This extends the theory of one-point extensions to the categorical setting.

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 $\mathbfΛ:=\left[\begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix} ight]$. First, we prove that there is an equivalence $\Big( \mathsf{Mod}(\mathcal{T}), \mathbb{G}\mathsf{Mod}(\mathcal{U})\Big) \simeq \mathrm{Mod}(\mathbfΛ)$. 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 $\mathbfΛ:=\left[\begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix} ight]$ is a dualizing variety (see theorem 6.10). In particular, $\mathrm{mod}(\mathbfΛ)$ 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.

Motivation & Objective

• To generalize triangular matrix algebras to the context of rings with several objects using additive categories.
• To define a triangular matrix category $\mathbf{\Lambda} = \left[\begin{smallmatrix}\mathcal{T}&0\\ M&\mathcal{U}\end{smallmatrix}\right]$ for additive categories $\mathcal{U}, \mathcal{T}$ and a bimodule $M \in \mathsf{Mod}(\mathcal{U} \otimes \mathcal{T}^{op})$.
• To establish conditions under which $\mathbf{\Lambda}$ becomes a dualizing variety, ensuring homological duality and the existence of Auslander–Reiten sequences.
• To generalize the classical one-point extension construction in representation theory to the categorical and functorial framework.

Proposed method

• Construct the triangular matrix category $\mathbf{\Lambda}$ as a comma-like category of triples $(A, B, f)$ with $f: M \otimes_{\mathcal{T}} A \to B$ a morphism in $\mathcal{U}$-modules.
• Establish an equivalence $\Big{(}\mathsf{Mod}(\mathcal{T}), \mathbb{G}\mathsf{Mod}(\mathcal{U})\Big{)} \simeq \mathrm{Mod}(\mathbf{\Lambda})$, linking module categories over $\mathcal{T}$ and $\mathcal{U}$ to modules over $\mathbf{\Lambda}$.
• Use the notion of dualizing $K$-varieties, requiring duality functors between $\mathrm{mod}(\mathcal{C})$ and $\mathrm{mod}(\mathcal{C}^{op})$ for a category $\mathcal{C}$.
• Apply homological tools such as derived functors $L^n$ and Ext functors to verify that $\mathrm{Ext}^n_{\mathcal{C}}(-,C)$ and $\mathrm{Ext}^n_{\mathcal{C}}(C,-)$ are finitely presented, ensuring the dualizing property.
• Prove that the triangular matrix category $\left[\begin{smallmatrix}\mathcal{C}&0\\ \widehat{\mathbbm{Hom}}\\ \mathcal{C}\end{smallmatrix}\right]$ is dualizing when $\mathcal{C}$ is a dualizing variety.
• Use the derived functor isomorphism $\mathbb{D}_{\mathcal{C}^{op}}\overline{\mathbb{F}}(B) \simeq L^n(\mathbb{D}_{\mathcal{C}^{op}}(B))$ to verify duality in the matrix category.

Experimental results

Research questions

• RQ1 Under what conditions is a triangular matrix category $\mathbf{\Lambda} = \left[\begin{smallmatrix}\mathcal{T}&0\\ M&\mathcal{U}\end{smallmatrix}\right]$ a dualizing variety when $\mathcal{U}$ and $\mathcal{T}$ are dualizing $K$-varieties?
• RQ2 How can the classical one-point extension construction in representation theory be generalized to the setting of additive categories and functor categories?
• RQ3 What role do derived functors $L^n$ and $\mathrm{Ext}^n$ play in ensuring the dualizing property of triangular matrix categories?
• RQ4 Can the category of maps (e.g., monomorphisms or epimorphisms) between modules be realized as a triangular matrix category with dualizing structure?
• RQ5 What is the relationship between the duality functors on $\mathcal{C}$ and the induced duality on the matrix category $\left[\begin{smallmatrix}\mathcal{C}&0\\ \widehat{\mathbbm{Ext}^n}\\ \mathcal{C}\end{smallmatrix}\right]$?

Key findings

• The triangular matrix category $\mathbf{\Lambda} = \left[\begin{smallmatrix}\mathcal{T}&0\\ M&\mathcal{U}\end{smallmatrix}\right]$ is a dualizing variety if $\mathcal{U}$ and $\mathcal{T}$ are dualizing $K$-varieties and $M$ satisfies specific finiteness and representability conditions.
• The equivalence $\Big{(}\mathsf{Mod}(\mathcal{T}), \mathbb{G}\mathsf{Mod}(\mathcal{U})\Big{)} \simeq \mathrm{Mod}(\mathbf{\Lambda})$ holds, generalizing module categories over matrix rings.
• If $\mathcal{C}$ is a dualizing $K$-variety, then $\left[\begin{smallmatrix}\mathcal{C}&0\\ \widehat{\mathbbm{Hom}}\\ \mathcal{C}\end{smallmatrix}\right]$ is a dualizing variety.
• For a dualizing $K$-variety $\mathcal{C}$ with enough projectives, $\left[\begin{smallmatrix}\mathcal{C}&0\\ \widehat{\mathbbm{Ext}^n}\\ \mathcal{C}\end{smallmatrix}\right]$ is dualizing when $\mathrm{Ext}^n_{\mathcal{C}}(-,C)$ and $\mathrm{Ext}^n_{\mathcal{C}}(C,-)$ are finitely presented for all $C \in \mathcal{C}$.
• The duality on the matrix category satisfies $\mathbb{D}_{\mathcal{C}^{op}}\overline{\mathbb{F}}(B) \simeq L^n(\mathbb{D}_{\mathcal{C}^{op}}(B))$ if $\mathbb{D}_{\mathcal{C}^{op}}(B)$ is right exact.
• In particular, for $n=1$, the category $\left[\begin{smallmatrix}\mathcal{C}&0\\ \widehat{\mathbbm{Ext}^1}\\ \mathcal{C}\end{smallmatrix}\right]$ is dualizing, and the duality is compatible with the first derived functor.