| Title:
|
Wakamatsu tilting modules with finite injective dimension (English) |
| Author:
|
Zhao, Guoqiang |
| Author:
|
Yin, Lirong |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
4 |
| Year:
|
2013 |
| Pages:
|
865-876 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a left Noetherian ring, $S$ a right Noetherian ring and $_R\omega $ a Wakamatsu tilting module with $S={\rm End}(_R\omega )$. We introduce the notion of the $\omega $-torsionfree dimension of finitely generated $R$-modules and give some criteria for computing it. For any $n\geq 0$, we prove that ${\rm l.id}_R(\omega ) = {\rm r.id}_S(\omega )\leq n$ if and only if every finitely generated left $R$-module and every finitely generated right $S$-module have $\omega $-torsionfree dimension at most $n$, if and only if every finitely generated left $R$-module (or right $S$-module) has generalized Gorenstein dimension at most $n$. Then some examples and applications are given. (English) |
| Keyword:
|
Wakamatsu tilting module |
| Keyword:
|
$\omega $-$k$-torsionfree module |
| Keyword:
|
$\mathcal {X}$-resolution dimension |
| Keyword:
|
injective dimension |
| Keyword:
|
$\omega $-torsionless property |
| MSC:
|
16E10 |
| MSC:
|
16E30 |
| idZBL:
|
Zbl 1299.16011 |
| idMR:
|
MR3165501 |
| DOI:
|
10.1007/s10587-013-0058-5 |
| . |
| Date available:
|
2014-01-28T14:01:59Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143602 |
| . |
| Reference:
|
[1] Auslander, M., Bridger, M.: Stable Module Theory.Memoirs of the American Mathematical Society 94 AMS, Providence (1969). Zbl 0204.36402, MR 0269685, 10.1090/memo/0094 |
| Reference:
|
[2] Auslander, M., Buchweitz, R.-O.: The homological theory of maximal Cohen-Macaulay approximations.Mém. Soc. Math. Fr., Nouv. Sér. 38 (1989), 5-37. Zbl 0697.13005, MR 1044344 |
| Reference:
|
[3] Auslander, M., Reiten, I.: Cohen-Macaulay and Gorenstein Artin algebras.Representation Theory of Finite Groups and Finite-Dimensional Algebras G. O. Michler et al. Proc. Conf., Bielefeld/Ger. 1991, Prog. Math. 95 Birkhäuser, Basel (1991), 221-245. Zbl 0776.16003, MR 1112162 |
| Reference:
|
[4] Beligiannis, A., Reiten, I.: Homological and Homotopical Aspects of Torsion Theories.Memoirs of the American Mathematical Society 883 AMS, Providence (2007). Zbl 1124.18005, MR 2327478 |
| Reference:
|
[5] Bennis, D., Mahdou, N.: Global Gorenstein dimensions.Proc. Am. Math. Soc. 138 (2010), 461-465. Zbl 1205.16007, MR 2557164, 10.1090/S0002-9939-09-10099-0 |
| Reference:
|
[6] Göbel, R., Trlifaj, J.: Approximations and Endomorphism Algebras of Modules.De Gruyter Expositions in Mathematics 41 Walter de Gruyter, Berlin (2006). Zbl 1121.16002, MR 2251271 |
| Reference:
|
[7] Hoshino, M.: Algebras of finite self-injective dimension.Proc. Am. Math. Soc. 112 (1991), 619-622. Zbl 0737.16003, MR 1047011, 10.1090/S0002-9939-1991-1047011-8 |
| Reference:
|
[8] 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:
|
[9] Huang, Z.: Generalized tilting modules with finite injective dimension.J. Algebra 311 (2007), 619-634. Zbl 1130.16008, MR 2314727, 10.1016/j.jalgebra.2006.11.025 |
| Reference:
|
[10] Huang, Z.: Selforthogonal modules with finite injective dimension III.Algebr. Represent. Theory 12 (2009), 371-384. Zbl 1171.16006, MR 2501192, 10.1007/s10468-009-9157-2 |
| Reference:
|
[11] Huang, Z.: Wakamatsu tilting modules, $U$-dominant dimension, and $k$-Gorenstein modules.Abelian Groups, Rings, Modules, and Homological Algebra Lecture Notes in Pure and Applied Mathematics 249. Selected papers of a conference on the occasion of Edgar Earle Enochs' 72nd birthday, Auburn, AL, USA, September 9-11, 2004 P. Goeters et al. Chapman & Hall/CRC (2006), 183-202. Zbl 1102.16006, MR 2229112 |
| Reference:
|
[12] 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:
|
[13] Mantese, F., Reiten, I.: Wakamatsu tilting modules.J. Algebra 278 (2004), 532-552. Zbl 1075.16006, MR 2071651, 10.1016/j.jalgebra.2004.03.023 |
| Reference:
|
[14] Rotman, J. J.: An Introduction to Homological Algebra.Pure and Applied Mathematics 85 Academic Press, New York (1979). Zbl 0441.18018, MR 0538169 |
| Reference:
|
[15] Wakamatsu, T.: Tilting modules and Auslander's Gorenstein property.J. Algebra 275 (2004), 3-39. Zbl 1076.16006, MR 2047438, 10.1016/j.jalgebra.2003.12.008 |
| . |
