On the finiteness of the Gorenstein dimension for Artin algebras
Abstract
In SSZ, the authors proved that an Artin algebra A with infinite global dimension has an indecomposable module with infinite projective and infinite injective dimension, giving a new characterisation of algebras with finite global dimension. We prove in this article that an Artin algebra A that is not Gorenstein has an indecomposable A-module with infinite Gorenstein projective dimension and infinite Gorenstein injective dimension, which gives a new characterisation of algebras with finite Gorenstein dimension. We show that this gives a proper generalisation of the result in SSZ for Artin algebras.
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