2019/03/10 by Alicia León-Galeana, León-Galeana, Alicia, Martı́n Ortiz-Morales +3 · 2 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1903.03926
openalex publication_date 2019/03/10 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
In this paper we continue the study of triangular matrix categories \mathbfΛ=[ \beginsmallmatrix T & 0 M & U \endsmallmatrix] initiated in [21]. First, given an additive category C and an ideal IB in C, we prove a well known result that there is a canonical recollement \xymatrixMod(C/IB)\ar[r] & Mod(C)\ar[r]\ar@[l]\ar@<1ex>[l] & Mod(B)\ar@[l]\ar@<1ex>[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 C, we describe the maps category of mod(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 Smalø ([35, Theorem 2.1]), which give us a way of construct functorially finite subcategories in the category Mod([ \beginsmallmatrix T & 0 M & U \endsmallmatrix]) from those of Mod(T) and Mod(U).