Charles Paquette

A Non-Existence Theorem for Almost Split Sequences

Let k be a field, Q a quiver with countably many vertices and I an ideal of kQ such that kQ/I is a spectroid. In this note, we prove that there is no almost split sequence ending at an indecomposable not finitely presented representation of the bound quiver (Q, I). We then get that an indecomposable representation M of (Q, I) is the ending term of an almost split sequence if and only if it is finitely presented and not projective. The dual results are also true.

Strictly Stratified Algebras Revisited

In [1], Ágoston, Dlab, and Lukács introduced the notion of strictly stratified algebras. These algebras are stratified in the sense of Cline, Parshall, and Scott (see [5]) and contain the well-known class of standardly stratified algebras. The latter is well described from a homological point of view. Indeed, many homological conjectures such as the finitistic dimension conjectures (see [2]), the Cartan determinant conjecture (see [10]) and the strong no loop conjecture (see [9]) hold true for standardly stratified algebras. In this article, we shall try to extend these results to strictly stratified algebras. The key idea is to show that the filtration condition of a strictly stratified algebra behaves well with respect to the extension groups. As main results, we establish the finitistic injective dimension conjecture, verify the Cartan determinant conjecture and its converse, and prove a weaker version of the strong no loop conjecture for strictly stratified algebras.

ISOTROPIC SCHUR ROOTS

In this paper, we study the isotropic Schur roots of an acyclic quiver Q with n vertices. We study the perpendicular category Ad

Representation theory of strongly locally finite quivers

\mathcal{A}(d)

Semi-stable subcategories for Euclidean quivers

of a dimension vector d and give a complete description of it when d is an isotropic Schur δ. This is done by using exceptional sequences and by defining a subcategory ℛ(Q, δ) attached to the pair (Q, δ). The latter category is always equivalent to the category of representations of a connected acyclic quiver Qℛ of tame type, having a unique isotropic Schur root, say δℛ. The understanding of the simple objects in Aδ

Cluster categories of type \({\mathbb {A}}_\infty ^\infty \) and triangulations of the infinite strip

\mathcal{A}\left(\delta \right)