arXiv subjects
Publications and source records attributed to Bo Le.
An Artin algebra is called a composition factor algebra if its indecomposable finitely generated modules are determined, up to isomorphism, by their composition factors. We prove that every composition factor algebra is representation-finite. This settles a conjecture of Auslander, Reiten and Smalø. We also give criteria for comma categories and factorization categories to be of finite representation type.
We classify the connected components of support $τ$-tilting graphs of surface rigid algebras associated with partial tagged triangulations of punctured marked surfaces with nonempty boundary. For a connected algebra, the graph is disconnected if and only if the algebra is weakly symmetric, has at least two simple modules, and has Cartan rank one. In this case, it has exactly two connected components. For every nonzero $τ$-rigid pair over a connected surface rigid algebra, the graph of $τ$-tilting completions is connected. More generally, we classify the completion components for an arbitrary surface rigid algebra and a fixed $τ$-rigid pair. For partial tagged triangulations with all endpoints at punctures, we express the Cartan matrix and $g$-vectors by signed endpoint data. We prove that $τ$-tilting finiteness is equivalent to positive definiteness of the Cartan matrix. In the finite case, we determine the primitive $g$-vector ray generators in endpoint coordinates. Trees give type A, and a criterion on the original partial tagged triangulation distinguishes types D and C in the unicyclic case.
We give a new equivalent formulation of Verdier's octahedral axiom. An initial square in a pre-triangulated category determines an obstruction in a quotient of ordinary Hom groups. Its vanishing is equivalent to the existence of a good completion, and $\mathrm{TR4}$ is equivalent to this vanishing for the squares associated with composable morphisms. As a consequence, if two pre-triangulations have the same underlying additive category and one is triangulated, then the other satisfies $\mathrm{TR4}$ precisely when their relative inverse Heller comparison admits an exact lift along every short exact sequence in the Freyd category. As applications, we prove that the exotic pre-triangulated category of Díaz Cabrera--Muro is in fact a triangulated category over every algebraically closed field. We construct scalar families for type-$A_{3a-1}$ preprojective algebras over arbitrary fields and for a range of Dynkin preprojective algebras in characteristic two; in each family the zero parameter is the unique triangulated member. The local equations also yield separable descent, including canonical triangulations for non-modular equivariantizations and for the Markman--Mehrotra K3 deformation categories.
In this paper, we associate a quiver with superpotential to each $d$-angulation of a (unpunctured) marked surface. We show that, under quasi-isomorphisms, the flip of a $d$-angulation is compatible with Oppermann's mutation of (the Ginzburg algebra of) the corresponding quiver with superpotential, thereby partially generalizing the result in [LF09]. Applying to the generalized $(d-2)$-cluster categories associated to this quiver with superpotential, we prove that some certain almost complete $(d-2)$-cluster tilting objects in the higher cluster category have exactly $d-1$ complements.