Triangular matrix categories II: Recollements and functorially finite subcategories

Abstract

In this paper we continue the study of triangular matrix categories $\mathbf {{\varLambda }}=\left [\begin {smallmatrix} \mathcal {T} & 0 \\ M & \mathcal {U} \end {smallmatrix}\right ]$ initiated in León-Galeana et al. (2022). First, given a additive category $\mathcal {C}$ and an ideal $\mathcal {I}_{{\mathscr{B}}}$ in $\mathcal {C}$, we prove a well known result that there is a canonical recollement

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 (J. Algebra, 321 (9), 2474–2485 2009, [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 $\text {mod}(\mathcal {C})$ 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 Martínez-Villa and Ortíz-Morales (Inter. J Algebra, 5 (11), 529–561 2011). Finally, we prove a generalization of a result due to Smalø (2011, [Theorem 2.1]), which give us a way of construct functorially finite subcategories in the category $\text {Mod}\Big (\left [\begin {smallmatrix} \mathcal {T} & 0 \\ M & \mathcal {U} \end {smallmatrix}\right ]\Big )$ from those of $\text {Mod}(\mathcal {T})$ and $\text {Mod}(\mathcal {U})$.