Wakamatsu tilting modules with finite injective dimension

Guoqiang Zhao; Lirong Yin

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 4, page 865-876
  • ISSN: 0011-4642

Abstract

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Let R be a left Noetherian ring, S a right Noetherian ring and R ω a Wakamatsu tilting module with S = End ( R ω ) . We introduce the notion of the ω -torsionfree dimension of finitely generated R -modules and give some criteria for computing it. For any n ≥ 0 , we prove that l . id R ( ω ) = r . id S ( ω ) ≤ n if and only if every finitely generated left R -module and every finitely generated right S -module have ω -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.

How to cite

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Zhao, Guoqiang, and Yin, Lirong. "Wakamatsu tilting modules with finite injective dimension." Czechoslovak Mathematical Journal 63.4 (2013): 865-876. <http://eudml.org/doc/260786>.

@article{Zhao2013,
abstract = {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 $n\ge 0$, we prove that $\{\rm l.id\}_R(\omega ) = \{\rm r.id\}_S(\omega )\le n$ 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.},
author = {Zhao, Guoqiang, Yin, Lirong},
journal = {Czechoslovak Mathematical Journal},
keywords = {Wakamatsu tilting module; $\omega $-$k$-torsionfree module; $\mathcal \{X\}$-resolution dimension; injective dimension; $\omega $-torsionless property; Wakamatsu tilting modules; injective dimension; generalized tilting modules; torsionfree dimension; finitely generated left modules; generalized Gorenstein dimension},
language = {eng},
number = {4},
pages = {865-876},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Wakamatsu tilting modules with finite injective dimension},
url = {http://eudml.org/doc/260786},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Zhao, Guoqiang
AU - Yin, Lirong
TI - Wakamatsu tilting modules with finite injective dimension
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 4
SP - 865
EP - 876
AB - 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 $n\ge 0$, we prove that ${\rm l.id}_R(\omega ) = {\rm r.id}_S(\omega )\le n$ 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.
LA - eng
KW - Wakamatsu tilting module; $\omega $-$k$-torsionfree module; $\mathcal {X}$-resolution dimension; injective dimension; $\omega $-torsionless property; Wakamatsu tilting modules; injective dimension; generalized tilting modules; torsionfree dimension; finitely generated left modules; generalized Gorenstein dimension
UR - http://eudml.org/doc/260786
ER -

References

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  1. Auslander, M., Bridger, M., 10.1090/memo/0094, Memoirs of the American Mathematical Society 94 AMS, Providence (1969). (1969) Zbl0204.36402MR0269685DOI10.1090/memo/0094
  2. Auslander, M., Buchweitz, R.-O., The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. Fr., Nouv. Sér. 38 (1989), 5-37. (1989) Zbl0697.13005MR1044344
  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. (1991) Zbl0776.16003MR1112162
  4. Beligiannis, A., Reiten, I., Homological and Homotopical Aspects of Torsion Theories, Memoirs of the American Mathematical Society 883 AMS, Providence (2007). (2007) Zbl1124.18005MR2327478
  5. Bennis, D., Mahdou, N., 10.1090/S0002-9939-09-10099-0, Proc. Am. Math. Soc. 138 (2010), 461-465. (2010) Zbl1205.16007MR2557164DOI10.1090/S0002-9939-09-10099-0
  6. Göbel, R., Trlifaj, J., Approximations and Endomorphism Algebras of Modules, De Gruyter Expositions in Mathematics 41 Walter de Gruyter, Berlin (2006). (2006) Zbl1121.16002MR2251271
  7. Hoshino, M., 10.1090/S0002-9939-1991-1047011-8, Proc. Am. Math. Soc. 112 (1991), 619-622. (1991) Zbl0737.16003MR1047011DOI10.1090/S0002-9939-1991-1047011-8
  8. Huang, Z., 10.1007/BF02874420, Sci. China, Ser. A 44 (2001), 184-192. (2001) Zbl1054.16002MR1824318DOI10.1007/BF02874420
  9. Huang, Z., 10.1016/j.jalgebra.2006.11.025, J. Algebra 311 (2007), 619-634. (2007) Zbl1130.16008MR2314727DOI10.1016/j.jalgebra.2006.11.025
  10. Huang, Z., 10.1007/s10468-009-9157-2, Algebr. Represent. Theory 12 (2009), 371-384. (2009) Zbl1171.16006MR2501192DOI10.1007/s10468-009-9157-2
  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. (2006) Zbl1102.16006MR2229112
  12. 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
  13. Mantese, F., Reiten, I., 10.1016/j.jalgebra.2004.03.023, J. Algebra 278 (2004), 532-552. (2004) Zbl1075.16006MR2071651DOI10.1016/j.jalgebra.2004.03.023
  14. Rotman, J. J., An Introduction to Homological Algebra, Pure and Applied Mathematics 85 Academic Press, New York (1979). (1979) Zbl0441.18018MR0538169
  15. Wakamatsu, T., 10.1016/j.jalgebra.2003.12.008, J. Algebra 275 (2004), 3-39. (2004) Zbl1076.16006MR2047438DOI10.1016/j.jalgebra.2003.12.008

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