Abstract
Let A be a finite dimensional algebra over a field K with enveloping algebra Ae=Aop⊗KA. Following [13], we call algebras A that have the property that the subcategory of Gorenstein projective modules in mod−A coincides with the subcategory {X∈mod−A|ExtAi(X,A)=0 for all i≥1} left weakly Gorenstein. The class of left weakly Gorenstein algebras is a large class that includes for example all Gorenstein algebras and all representation-finite algebras. We prove that the Gorenstein dimension of A coincides with the Gorenstein projective dimension of the regular module as an Ae-module for left weakly Gorenstein algebras A. We give three application of this result. The first generalises a formula by Happel for the global dimension of algebras. The second application generalises a criterion of Shen for an algebra to be selfinjective. As a final application we prove a stronger version of the first Tachikawa conjecture for left weakly Gorenstein algebras.
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