On string algebras and the Cohen-Macaulay Auslander algebras (Q6631555)

This paper provides a clear and insightful treatment of the Cohen-Macaulay Auslander algebra of an algebra \(A\), specifically focusing on the case of string algebras. The definition of the Cohen-Macaulay Auslander algebra as the endomorphism algebra of the direct sum of all indecomposable Gorenstein projective \(A\)-modules is well explained, and the explicit construction of this algebra for string algebras is a valuable contribution to the field. Furthermore, the paper introduces a class of string algebras satisfying the \(G\)-condition and proves a key result: these algebras are representation-finite if and only if their Cohen-Macaulay Auslander algebras are also representation-finite. This result highlights the deep interplay between the representation theory of string algebras and their Cohen-Macaulay Auslander algebras, offering a new perspective on the conditions for representation-finiteness in this context.\N\NAs an important application, the paper proves that the derived representation type of gentle algebras coincides with that of their Cohen-Macaulay Auslander algebras, shedding light on the derived category structure of these algebras. Overall, the paper makes a significant contribution to the study of Cohen-Macaulay Auslander algebras, string algebras, and their representation theory, with clear proofs and important results that will be of interest to researchers in algebra and representation theory.