On a new formula for the Gorenstein dimension

JOURNAL OF ALGEBRA

卷:526

Article

Marczinzik, Rene¹ 

[1] Univ Stuttgart, Inst Algebra & Number Theory, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

关键词: Gorenstein algebras;    Enveloping algebra;    Gorenstein homological algebra;    Gorenstein projective dimension;   

DOI  :  10.1016/j.jalgebra.2019.02.013

来源: Elsevier

【 摘 要 】

Let A be a finite dimensional algebra over a field K with enveloping algebra A(e) = A(op) circle times(K) A. 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 is an element of mod - A vertical bar Ext(A)(i)(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 A(e)-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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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