Czechoslovak Mathematical Journal, Vol. 63, No. 1, pp. 143-156, 2013

A generalization of the Auslander transpose and the generalized Gorenstein dimension

Yuxian Geng

Yuxian Geng, School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China, e-mail:

Abstract: Let $R$ be a left and right Noetherian ring and $C$ a semidualizing $R$-bimodule. We introduce a transpose $ Tr_cM$ of an $R$-module $M$ with respect to $C$ which unifies the Auslander transpose and Huang's transpose, see Z. Y. Huang, On a generalization of the Auslander-Bridger transpose, Comm. Algebra 27 (1999), 5791-5812, in the two-sided Noetherian setting, and use $ Tr_cM$ to develop further the generalized Gorenstein dimension with respect to $C$. Especially, we generalize the Auslander-Bridger formula to the generalized Gorenstein dimension case. These results extend the corresponding ones on the Gorenstein dimension obtained by Auslander in M. Auslander, M. Bridger, Stable Module Theory, Mem. Amer. Math. Soc. vol. 94, Amer. Math. Soc., Providence, RI, 1969.

Keywords: transpose, semidualizing module, generalized Gorenstein dimension, depth, Auslander-Bridger formula

Classification (MSC 2010): 13C15, 13E05, 16E10, 16P40

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