| Title:
|
A generalization of the Auslander transpose and the generalized Gorenstein dimension (English) |
| Author:
|
Geng, Yuxian |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
143-156 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
transpose |
| Keyword:
|
semidualizing module |
| Keyword:
|
generalized Gorenstein dimension |
| Keyword:
|
depth |
| Keyword:
|
Auslander-Bridger formula |
| MSC:
|
13C15 |
| MSC:
|
13E05 |
| MSC:
|
16E10 |
| MSC:
|
16P40 |
| idZBL:
|
Zbl 1274.13022 |
| idMR:
|
MR3035502 |
| DOI:
|
10.1007/s10587-013-0009-1 |
| . |
| Date available:
|
2013-03-01T16:10:36Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143175 |
| . |
| Reference:
|
[1] Auslander, M., Bridger, M.: Stable Module Theory.Mem. Am. Math. Soc. 94 (1969). Zbl 0204.36402, MR 0269685 |
| Reference:
|
[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. Zbl 0776.16003, MR 1112162 |
| Reference:
|
[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). MR 0979760 |
| Reference:
|
[4] Cartan, H., Eilenberg, S.: Homological Algebra.Princeton Mathematical Series, 19 Princeton University Press XV (1956). Zbl 0075.24305, MR 0077480 |
| Reference:
|
[5] Christensen, L. W.: Gorenstein Dimension.Lecture Notes in Mathematics 1747 Springer, Berlin (2000). MR 1799866, 10.1007/BFb0103984 |
| Reference:
|
[6] Christensen, L. W.: Semi-dualizing complexes and their Auslander categories.Trans. Am. Math. Soc. 353 (2001), 1839-1883. Zbl 0969.13006, MR 1813596, 10.1090/S0002-9947-01-02627-7 |
| Reference:
|
[7] Enochs, E. E., Jenda, O. M. G.: Gorenstein injective and projective modules.Math. Z. 220 (1995), 611-633. Zbl 0845.16005, MR 1363858, 10.1007/BF02572634 |
| Reference:
|
[8] Foxby, H.-B.: Gorenstein modules and related modules.Math. Scand. 31 (1972), 276-284. MR 0327752 |
| Reference:
|
[9] Holm, H., Jørgensen, P.: Semi-dualizing modules and related Gorenstein homological dimensions.J. Pure Appl. Algebra 205 (2006), 423-445. MR 2203625, 10.1016/j.jpaa.2005.07.010 |
| Reference:
|
[10] Holm, H., White, D.: Foxby equivalence over associative rings.J. Math. Kyoto Univ. 47 (2007), 781-808. Zbl 1154.16007, MR 2413065, 10.1215/kjm/1250692289 |
| Reference:
|
[11] Huang, Z.: On a generalization of the Auslander-Bridger transpose.Commun. Algebra 27 (1999), 5791-5812. Zbl 0948.16007, MR 1726277, 10.1080/00927879908826791 |
| Reference:
|
[12] Huang, Z.: $\omega$-$k$-torsionfree modules and $\omega$-left approximation dimension.Sci. China, Ser. A 44 (2001), 184-192. Zbl 1054.16002, MR 1824318, 10.1007/BF02874420 |
| Reference:
|
[13] Huang, Z., Tang, G.: Self-orthogonal modules over coherent rings.J. Pure Appl. Algebra 161 (2001), 167-176. Zbl 0989.16005, MR 1834083, 10.1016/S0022-4049(00)00109-2 |
| Reference:
|
[14] Matsumura, H.: Commutative Algebra. 2nd ed.Mathematics Lecture Note Series, 56 The Benjamin/Cummings Publishing Company, Reading, Massachusetts (1980). Zbl 0441.13001, MR 0575344 |
| Reference:
|
[15] Strooker, J. R.: An Auslander-Buchsbaum identity for semidualizing modules.Available from the arXiv: math.AC/0611643. |
| Reference:
|
[16] Wakamatsu, T.: On modules with trivial self-extensions.J. Algebra 114 (1988), 106-114. Zbl 0646.16025, MR 0931903, 10.1016/0021-8693(88)90215-3 |
| Reference:
|
[17] White, D.: Gorenstein projective dimension with respect to a semidualizing module.J. Commut. Algebra 2 (2010), 111-137. MR 2607104, 10.1216/JCA-2010-2-1-111 |
| . |
