Cycle-Finite Module Categories

Abstract

We describe the structure of module categories of finite dimensional algebras over an algebraically closed field for which the cycles of nonzero nonisomorphisms between indecomposable finite dimensional modules are finite (do not belong to the infinite Jacobson radical of the module category). Moreover, geometric and homological properties of these module categories are exhibited.

The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

“…We know from [40, Theorem 4.1] that A is cycle-finite and A admits a sincere stable tube if and only if A is either tame concealed or tubular algebra. Applying now the description of the structure of the Auslander-Reiten quiver of a tame generalized multicoil algebra [25,Theorem 4.8] (that is, a tame generalized multicoil enlargement of a finite family of tame concealed algebras), we conclude that any such algebra is cycle-finite. Using additionally [27, Theorem A] in the tame case, we receive that A is tame and A admits a separating family of almost cyclic coherent components if and only if A is cyclefinite and A admits a separating family of almost cyclic coherent components.…”

“…Further, (15,13) and (16,6) are related pairs such that there is the path 16 → 15 in Q, and the vertices 13, 6 belong to the same coextension branch and we have the path 6 → 13 in it. Moreover, the pairs (28,25) and (16,6) are not related nor do the pairs (28,25), (17,10) and (28,25), (15,13).…”

Auslander–Reiten components with bounded short cycles

“…Motivated by this fact, Skowroński studied in [26] cycles of maps of finite depth, which are originally called finite cycles. Indeed, Auslander-Reiten components whose non-directing modules lie only on cycles of maps of finite depth are described in [17], and algebras whose module category contains only cycles of maps of finite depth are extensively studied by many authors; see, for example, [16,17,23,26]. More generally, the connected components Γ of Γ A such that rad(Γ ) contains only short cycles of maps of finite depth are studied in [13].…”

Krull Dimension of Tame Generalized Multicoil Algebras

“…Moreover, recently the tame generalized multicoil algebras showed to be important in describing the structure of the module category ind of an arbitrary cycle-finite algebra (see [18,Theorems 7.1 and 7.2] or [19,Theorem 1.8]). We also refer to the article [24] for the Hochschild cohomology of generalized multicoil algebras.…”