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

top
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

top

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

top
  1. Auslander, M., Bridger, M., Stable Module Theory, Mem. Am. Math. Soc. 94 (1969). (1969) Zbl0204.36402MR0269685
  2. Auslander, M., Reiten, I., Cohen-Macaulay and Gorenstein Artin algebras, Representation Theory of Finite Groups and Finite-Dimensional Algebras, Proc. Conf., Bielefeld/Ger. Prog. Math. 95, Birkhäuser, Basel (1991), 221-245. (1991) Zbl0776.16003MR1112162
  3. Bourbaki, N., Elements of Mathematics. Commutative Algebra. Chapters 1-7. Transl. from the French. Softcover Edition of the 2nd printing 1989, Springer, Berlin (1989). (1989) MR0979760
  4. Cartan, H., Eilenberg, S., Homological Algebra, Princeton Mathematical Series, 19 Princeton University Press XV (1956). (1956) Zbl0075.24305MR0077480
  5. Christensen, L. W., 10.1007/BFb0103984, Lecture Notes in Mathematics 1747 Springer, Berlin (2000). (2000) MR1799866DOI10.1007/BFb0103984
  6. Christensen, L. W., 10.1090/S0002-9947-01-02627-7, Trans. Am. Math. Soc. 353 (2001), 1839-1883. (2001) Zbl0969.13006MR1813596DOI10.1090/S0002-9947-01-02627-7
  7. Enochs, E. E., Jenda, O. M. G., 10.1007/BF02572634, Math. Z. 220 (1995), 611-633. (1995) Zbl0845.16005MR1363858DOI10.1007/BF02572634
  8. Foxby, H.-B., Gorenstein modules and related modules, Math. Scand. 31 (1972), 276-284. (1972) MR0327752
  9. Holm, H., Jørgensen, P., 10.1016/j.jpaa.2005.07.010, J. Pure Appl. Algebra 205 (2006), 423-445. (2006) MR2203625DOI10.1016/j.jpaa.2005.07.010
  10. Holm, H., White, D., 10.1215/kjm/1250692289, J. Math. Kyoto Univ. 47 (2007), 781-808. (2007) Zbl1154.16007MR2413065DOI10.1215/kjm/1250692289
  11. Huang, Z., 10.1080/00927879908826791, Commun. Algebra 27 (1999), 5791-5812. (1999) Zbl0948.16007MR1726277DOI10.1080/00927879908826791
  12. Huang, Z., 10.1007/BF02874420, Sci. China, Ser. A 44 (2001), 184-192. (2001) Zbl1054.16002MR1824318DOI10.1007/BF02874420
  13. Huang, Z., Tang, G., 10.1016/S0022-4049(00)00109-2, J. Pure Appl. Algebra 161 (2001), 167-176. (2001) Zbl0989.16005MR1834083DOI10.1016/S0022-4049(00)00109-2
  14. Matsumura, H., Commutative Algebra. 2nd ed, Mathematics Lecture Note Series, 56 The Benjamin/Cummings Publishing Company, Reading, Massachusetts (1980). (1980) Zbl0441.13001MR0575344
  15. Strooker, J. R., An Auslander-Buchsbaum identity for semidualizing modules, Available from the arXiv: math.AC/0611643. 
  16. Wakamatsu, T., 10.1016/0021-8693(88)90215-3, J. Algebra 114 (1988), 106-114. (1988) Zbl0646.16025MR0931903DOI10.1016/0021-8693(88)90215-3
  17. White, D., 10.1216/JCA-2010-2-1-111, J. Commut. Algebra 2 (2010), 111-137. (2010) MR2607104DOI10.1216/JCA-2010-2-1-111

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.