Indecomposable pure-injective objects in stable categories of Gorenstein-projective modules over Gorenstein orders

TL;DR

This paper characterizes when a complete Gorenstein order has finite Cohen-Macaulay representation type by examining indecomposable pure-injective objects in the stable category of Gorenstein-projective modules, extending classical results.

Contribution

It establishes a new criterion linking pure-injective objects to the representation type of Gorenstein orders, generalizing Auslander-Ringel-Tachikawa results.

Findings

Finite Cohen-Macaulay representation type iff all indecomposable pure-injective objects are compact

Provides a characterization of Gorenstein orders via stable category properties

Extends classical representation theory results to Gorenstein contexts

Abstract

We give a result of Auslander-Ringel-Tachikawa type for Gorenstein-projective modules over a complete Gorenstein order. In particular, we prove that a complete Gorenstein order is of finite Cohen-Macaulay representation type if and only if every indecomposable pure-injective object in the stable category of Gorenstein-projective modules is compact.