Indecomposable modules over multicoil algebras.

*(English)*Zbl 0796.16014The authors generalize the notion of coil-algebras [introduced in Manuscr. Math. 67, No. 3, 305-331 (1990; Zbl 0696.16023)] and initiate the study of the so-called multicoil algebras. Let \(k\) be an algebraically closed field and \(A\) a finite-dimensional, basic and connected \(k\)-algebra. Moreover let \(\bmod A\) be the category of finitely generated right \(A\)-modules and \(\Gamma_ A\) the Auslander-Reiten quiver of \(A\). A coil is, roughly speaking, a translation quiver which is obtained from a stable tube by some admissible operations. A component \(\Gamma\) of \(\Gamma_ A\) is called a multicoil if it contains a full subquiver \(\Gamma'\) which is a finite union of standard coils and has the property that the indecomposables not belonging to \(\Gamma'\) are directing in \(\bmod A\). Finally, \(A\) is said to be a multicoil algebra if any cycle of non-zero non-isomorphisms \(M = M_ 0 \to M_ 1 \to \dots \to M_ t = M\) in \(\bmod A\) lies in one multicoil of \(\Gamma_ A\). In particular multicoil algebras are cycle-finite and hence tame.

The main result proved in the paper is that for a multicoil algebra \(A\) the following conditions are equivalent: (i) \(A\) is either tame concealed or tubular, (ii) there exists a sincere indecomposable \(A\)-module lying in a stable tube of \(\Gamma_ A\), (iii) there exist infinitely many non- isomorphic sincere modules of the same dimension type lying in homogeneous tubes of \(\Gamma_ A\).

It follows that the indecomposable modules lying in a stable tube of a multicoil algebra have as their support a tame concealed or a tubular full convex subcategory of the algebra. Moreover, it is proved that a multicoil algebra is of polynomial growth. As a consequence, a multicoil algebra is domestic if and only if it does not contain a tubular algebra as a full convex subcategory.

The main result proved in the paper is that for a multicoil algebra \(A\) the following conditions are equivalent: (i) \(A\) is either tame concealed or tubular, (ii) there exists a sincere indecomposable \(A\)-module lying in a stable tube of \(\Gamma_ A\), (iii) there exist infinitely many non- isomorphic sincere modules of the same dimension type lying in homogeneous tubes of \(\Gamma_ A\).

It follows that the indecomposable modules lying in a stable tube of a multicoil algebra have as their support a tame concealed or a tubular full convex subcategory of the algebra. Moreover, it is proved that a multicoil algebra is of polynomial growth. As a consequence, a multicoil algebra is domestic if and only if it does not contain a tubular algebra as a full convex subcategory.

Reviewer: H.Meltzer (Paderborn)

##### MSC:

16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |

16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |