On a new formula for the Gorenstein dimension

2017/10/09 by Marczinzik, Rene · 1 citation
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1710.03066

Abstract

Let A be a finite dimensional algebra over a field K with enveloping algebra A^e=A^op ⊗_K A. We call algebras A that have the property that the subcategory of Gorenstein projective modules in mod-A coincide with the subcategory \ X ∈ mod-A | Ext_Aⁱ(X,A)=0 for all i ≥ 1 \ left nearly Gorenstein. The class of left nearly 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 A^e-module for left nearly 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 nearly Gorenstein algebras.

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