Almost cyclic coherent components of an Auslander-Reiten quiver
The authors study very closely the cyclic part \(_c\Gamma_A\) of the A-R quiver of an Artin algebra \(A\). The definition of the quiver \(_c\Gamma_A\) is the following: A vertex \(X\) of the A-R quiver is called cyclic if it lies on an oriented cycle of \(\Gamma_A\). The cyclic part \(_c\Gamma_A\), of the A-R quiver, is the full translation subquiver of \(\Gamma_A\) whose vertices are the cyclic vertices. The connected components of the cyclic part are called the cyclic components of \(\Gamma_A\). One of the theorems states that two vertices are in the same cyclic component if and only if they lie on a common oriented cycle. The authors are interested in studying the cyclic coherent components. Recall that a translation subquiver \(\Gamma\) of \(\Gamma_A\) is called coherent if it satisfies the following two conditions: (C1) For each projective \(P\) in \(\Gamma\) there is an infinite sectional path \(P=X_1\to X_2\to\cdots\to X_i\to X_{i+1}\to X_{i+2}\to\cdots\) in \(\Gamma\), called a coray ending in \(P\). (C2) A similar condition has to be satisfied for injective vertices. A full translation subquiver \(\Gamma\) of \(\Gamma_A\) is said to be almost cyclic if all but a finite number of vertices of \(\Gamma\) lie on oriented cycles contained entirely in \(\Gamma\). These concepts are deeply related with the notion of coil, introduced by Assem and Skowroński, which was generalized to the notion generalized coil, by Skowroński, and now they introduce in this paper the notion of generalized multicoil. These notions are all defined in terms of a general translation quiver. I state now the most important theorems of the paper, (stated in the introduction of the paper): Theorem A. Let \(\Gamma\) be a connected component of \(\Gamma_A\). Then \(\Gamma\) is coherent and almost cyclic if and only if \(\Gamma\) is a generalized multicoil. Theorem B gets, as conclusion of Theorem A, the local structure of a coherent and almost cyclic connected component, (i.e. a generalized multicoil). Theorem C concludes that in a coherent and almost cyclic component there is exactly one infinite sectional path starting in each projective. Theorem D states the following: A connected component is a coil if and only if it is coherent, almost cyclic, and the fundamental group is infinite cyclic. Theorem E states that given a quiver \(\Gamma\) which is a generalized multicoil, there is always an Artin algebra which has \(\Gamma\) as connected component. Theorem F gives a combinatorial characterization of generalized multicoils. This review is just taken from the introduction of the paper, which (as the whole paper) is very good in the opinion of this reviewer. The proofs and the more technical definitions as for example the generalized multicoil appear after the introduction.
scientific article; zbMATH DE number 1502172
scientific article