On 1-Gorenstein Algebras of Finite Cohen–Macaulay Type

Abstract

An Artin algebra Λ is said to be of finite Cohen–Macaulay type if, up to isomorphism, there are only finitely many indecomposable modules in G(Λ), the full subcategory of modΛ consisting of all Gorenstein projective (right) Λ-modules. In this paper, we study 1-Gorenstein algebras of finite Cohen–Macaulay type through mod(G(Λ)), the category of finitely presented G(Λ)-modules. Some applications will be provided. In particular, a necessary and sufficient condition is given for T3(Λ), the 3 by 3 lower triangular matrices over Λ, to be of finite Cohen–Macaulay type. Finally, the structure of almost split sequences will be described explicitly in a special subcategory of mod(G(Λ)), denoted by ϑ−1(G(Λ)). If Λ is self-injective, ϑ−1(G(Λ))=mod(G(Λ)).