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 $\mathcal{G}(\mathrm{\Lambda })$, the full subcategory of $\mathrm{mod}\mathrm{\Lambda }$ consisting of all Gorenstein projective (right) Λ-modules. In this paper, we study 1-Gorenstein algebras of finite Cohen–Macaulay type through $\mathrm{mod}(\mathcal{G}(\mathrm{\Lambda }))$, the category of finitely presented $\mathcal{G}(\mathrm{\Lambda })$-modules. Some applications will be provided. In particular, a necessary and sufficient condition is given for ${\mathit{T}}_3(\mathrm{\Lambda })$, 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 $\mathrm{mod}(\mathcal{G}(\mathrm{\Lambda }))$, denoted by ${\mathit{\vartheta }}^{-1}(\mathcal{G}(\mathrm{\Lambda }))$. If Λ is self-injective, ${\mathit{\vartheta }}^{-1}(\mathcal{G}(\mathrm{\Lambda }))=\mathrm{mod}(\mathcal{G}(\mathrm{\Lambda }))$.