Triangular Matrix Categories II: Recollements and functorially finite subcategories

• Published in: arXiv.org 2019-03
• Main Authors: León-Galeana, Alicia, Ortiz-Morales, Martín, Valente Santiago Vargas
• Format: Article
• Language: English
• Subjects: Categories; Theorems
• ISSN: 2331-8422

In this paper we continue the study of triangular matrix categories \(\mathbf{\Lambda}=\left[ \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]\) initiated in [21]. First, given an additive category \(\mathcal{C}\) and an ideal \(\mathcal{I}_{\mathcal{B}}\) in \(\mathcal{C}\), we prove a well known result that there is a canonical recollement \(\xymatrix{\mathrm{Mod}(\mathcal{C}/\mathcal{I}_{\mathcal{B}})\ar[r]_{} & \mathrm{Mod}(\mathcal{C})\ar[r]_{}\ar@[l]_{}\ar@ [l]_{} & \mathrm{Mod}(\mathcal{B})\ar@[l]_{}\ar@ [l]_{}}\). We show that given a recollement between functor categories we can induce a new recollement between triangular matrix categories, this is a generalization of a result given by Chen and Zheng in [11, theorem 4.4]. In the case of dualizing \(K\)-varieties we can restrict the recollement we obtained to the categories of finitely presented functors. Given a dualizing variety \(\mathcal{C}\), we describe the maps category of \(\mathrm{mod}(\mathcal{C})\) as modules over a triangular matrix category and we study its Auslander-Reiten sequences and contravariantly finite subcategories, in particular we generalize several results from [24]. Finally, we prove a generalization of a result due to {Smal\o} ([35, Theorem 2.1]), which give us a way of construct functorially finite subcategories in the category \(\mathrm{Mod}\Big(\left[ \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]\Big)\) from those of \(\mathrm{Mod}(\mathcal{T})\) and \(\mathrm{Mod}(\mathcal{U})\).