2022/07/09 by Raziyeh Diyanatnezhad, Diyanatnezhad, Raziyeh, Alireza Nasr‐Isfahani +1
Mathematics · #Algebraic structures and combinatorial models #Rings, Modules, and Algebras #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2207.04225
Let Λ be an artin algebra and C be a functorially finite subcategory of modΛ which contains Λ or DΛ. We use the concept of the infinite radical of C and show that C has an additive generator if and only if rad^∞C vanishes. In this case we describe the morphisms in powers of the radical of C in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that C is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in C is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left C-approximations and right C-approximations of simple Λ-modules, we give other criteria to describe whether C is of finite representation type. In addition, we give a nilpotency index of the radical of C which is independent from the maximal length of indecomposable Λ-modules in C.