Given a finite dimensional algebra $ A$ . Define the homological index of a module $ M$ (all modules finite dimensional) as $ \psi(M):=pd(M)+id(M)+domdim(M)+codomdim(M)$ , that is the sum of the projective, injective, dominant and codominant dimension. In https://arxiv.org/abs/1608.04179 Iyama and Solberg introduced minimal Auslander-Gorenstein algebras as algebras having finite Gorenstein dimension $ g \geq 2$ equal to the dominant dimension, which generalise the famous class of higher Auslander algebras.
Now i can prove that the following are equivalent for an algebra A of finite Gorenstein dimension $ g \geq 2$ :
A is minimal Auslander-Gorenstein.
Each module $ M$ with finite homological index has homological index equal to $ 2g$ .
Questions: Is this still true in case we replace 2. by 2.’ where 2.’ states:”Each module $ M$ with finite homological index has constant homological index.”?
What is the corresponding result in case we do not restrict to Gorenstein algebras? That is: What are the algebras where all modules with finite index have their index equal to some constant $ c$ ?
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