On 1-Gorenstein Algebras of Finite Cohen-Macaulay Type

Abstract

An Artin algebra Λis 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 modΛconsisting 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 T3Λ, 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 g -1(g Λ). IfΛis self-injective, g -1(g Λ) = mod(g Λ). © 2023 University of Michigan. All rights reserved.