Algebras with separating almost cyclic coherent Auslander–Reiten components

Abstract

We prove that the class of Artin algebras whose Auslander-Reiten quiver admits a separating family of almost cyclic coherent components coincides with the class of generalized multicoil enlargements of concealed canonical algebras. Moreover, the module category, homological properties and the representation type of Artin algebras with separating families of almost cyclic coherent Auslander-Reiten components are described.

Krull Dimension of Tame Generalized Multicoil Algebras

“…Further, it was proved in [13] that the class of algebras with a separating family of stable tubes coincides with the class of concealed canonical algebras. This was extended in [21] to a characterization of algebras with a separating family of almost cyclic coherent Auslander-Reiten components. Recall that a component of an Auslander-Reiten quiver A is called almost cyclic if all but finitely many modules in lie on oriented cycles contained entirely in .…”

“…It has been proved in [21, Theorem A] that the Auslander-Reiten quiver A of an algebra A admits a separating family of almost cyclic coherent components if and only if A is a generalized multicoil enlargement of a (possibly decomposable) concealed canonical algebra C. Moreover, for such an algebra A, we have that A is triangular, gl.dim A ≤ 3, and pd A X ≤ 2 or id A X ≤ 2 for any module X in ind A (see [21,Corollary B and Theorem E] …”

“…This is the class of tame algebras among the class of all algebras having a separating family of almost cyclic coherent components investigated in [21,22]. Recall that a module X in mod A is called a brick if End A (X) ∼ = k.…”

“…For the operation (fad 1) s ≥ 0, for (fad 2) and (fad 4) s ≥ 1, and for (fad 3) s ≥ t − 1. In all above operations X s is injective (see [20] or [21] for the details).…”

“…If s = 1, such a translation quiver is said to be a generalized coil. The admissible operations of types (ad 1), (ad 2), (ad 3), (ad 1 * ), (ad 2 * ) and (ad 3 * ) have been introduced in [2][3][4], the admissible operations (ad 4) and (ad 4 * ) for r = 0 in [16], and the admissible operations (ad 4), (ad 4 * ) for r ≥ 1, (ad 5) and (ad 5 * ) in [20,21].…”