A generalization of the Auslander transpose and the generalized Gorenstein dimension

Yuxian Geng

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 1, page 143-156
  • ISSN: 0011-4642

Abstract

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Let R be a left and right Noetherian ring and C a semidualizing R -bimodule. We introduce a transpose Tr c M 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 c M 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.

How to cite

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Geng, Yuxian. "A generalization of the Auslander transpose and the generalized Gorenstein dimension." Czechoslovak Mathematical Journal 63.1 (2013): 143-156. <http://eudml.org/doc/252532>.

@article{Geng2013,
abstract = {Let $R$ be a left and right Noetherian ring and $C$ a semidualizing $R$-bimodule. We introduce a transpose $\{\rm Tr_\{c\}\}M$ 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 $\{\rm Tr_\{c\}\}M$ 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.},
author = {Geng, Yuxian},
journal = {Czechoslovak Mathematical Journal},
keywords = {transpose; semidualizing module; generalized Gorenstein dimension; depth; Auslander-Bridger formula; transpose; semidualizing module; generalized Gorenstein dimension; depth; Auslander-Bridger formula},
language = {eng},
number = {1},
pages = {143-156},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A generalization of the Auslander transpose and the generalized Gorenstein dimension},
url = {http://eudml.org/doc/252532},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Geng, Yuxian
TI - A generalization of the Auslander transpose and the generalized Gorenstein dimension
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 1
SP - 143
EP - 156
AB - Let $R$ be a left and right Noetherian ring and $C$ a semidualizing $R$-bimodule. We introduce a transpose ${\rm Tr_{c}}M$ 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 ${\rm Tr_{c}}M$ 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.
LA - eng
KW - transpose; semidualizing module; generalized Gorenstein dimension; depth; Auslander-Bridger formula; transpose; semidualizing module; generalized Gorenstein dimension; depth; Auslander-Bridger formula
UR - http://eudml.org/doc/252532
ER -

References

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