The existence totally reflexive covers

Document Type : Research Paper

Author

Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran.

10.29252/as.2019.1486

Abstract

Let $R$ be a commutative Noetherian ring. We prove that  over a local ring $R$ every finitely generated $R$-module $M$ of finite Gorenstein projective dimension has a Gorenstein projective cover
$\varphi:C \rightarrow M$ such that $C$ is finitely generated and the projective dimension of $\Ker\varphi$ is finite and $\varphi$ is surjective.

Keywords


[1] Christensen, L.W. Gorenstein dimensions, Lecture Notes in Mathematics, 1747. Springer, Berlin (2000).
[2] Christensen, L.W., Piepmeyer, G., Striuli, J., Takahashi, R. Finite Gorenstein representation type implies simple singularity, Adv.Math. 218, 1012-1026,(2008).
[3] Enochs, E.E., Jenda, O.M.G. Relative homological algebra, de Gruyter Expositions in Mathematics, 30 Walter de Gruyter and Co., Berlin (2000).
[4] Enochs, E.E., Jenda, O.M.G., Xu, J. A generalization of Auslanders last theorem, Algebr. Represent. Theory 2, 259-268 (1999).
[5] Holm, H. Gorenstein homological dimensions. J. Pure Appl. Algebra 189, 167-193 (2004).
[6] Takahashi, R. On the category of modules of Gorenstein dimension zero, Math. Z. 251, 249-256 (2005).