On a new formula for the Gorenstein dimension

TL;DR

This paper introduces a new formula linking Gorenstein dimension with Gorenstein projective dimension for a class of algebras called left nearly Gorenstein, with applications to global dimension, self-injectivity, and Tachikawa's conjecture.

Contribution

It establishes that for left nearly Gorenstein algebras, Gorenstein dimension equals the Gorenstein projective dimension of the regular module over the enveloping algebra, extending several classical results.

Findings

Gorenstein dimension equals Gorenstein projective dimension for left nearly Gorenstein algebras

Generalization of Happel's formula for global dimension

Enhanced criterion for algebra self-injectivity and Tachikawa's conjecture

Abstract

Let $A$ be a finite dimensional algebra over a field $K$ with enveloping algebra $A^e=A^{op} \otimes_K A$. We call algebras $A$ that have the property that the subcategory of Gorenstein projective modules in $mod-A$ coincide with the subcategory $\{ X \in mod-A | Ext_A^i(X,A)=0 $ for all $i \geq 1 \}$ 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.