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finitistic dimension; restricted injective dimension; tilting module
We study the relations between finitistic dimensions and restricted injective dimensions. Let $R$ be a ring and $T$ a left $R$-module with $A=\mathop {\rm End}_RT$. If $_RT$ is selforthogonal, then we show that $\mathop {\rm rid}(T_A)\leq \mathop {\rm findim}(A_A)\leq \mathop {\rm findim}(_RT)+\mathop {\rm rid}(T_A)$. Moreover, if $R$ is a left noetherian ring and $T$ is a finitely generated left \mbox {$R$-module} with finite injective dimension, then $\mathop {\rm rid}(T_A)\leq \mathop {\rm findim}(A_A)\leq \mathop {\rm fin.inj.dim}(_RR)+\mathop {\rm rid}(T_A)$. Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.
[1] Angeleri-Hügel, L., Trlifaj, J.: Tilting theory and the finitistic dimension conjectures. Trans. Am. Math. Soc. 354 (2002), 4345-4358. DOI 10.1090/S0002-9947-02-03066-0 | MR 1926879 | Zbl 1028.16004
[2] Asadollahi, J., Salarian, S.: Gorenstein injective dimension for complexes and Iwanaga-Gorenstein rings. Commun. Algebra 34 (2006), 3009-3022. DOI 10.1080/00927870600639815 | MR 2250584 | Zbl 1106.13024
[3] Auslander, M., Reiten, I., Smal{ø, S. O.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics 36 Cambridge University Press, Cambridge (1995). MR 1314422 | Zbl 0834.16001
[4] Buan, A. B., Krause, H., Solberg, {Ø.: On the lattice of cotilting modules. AMA, Algebra Montp. Announc. (electronic only) 2002 (2002), Paper 2, 6 pages. MR 1882670 | Zbl 1012.16014
[5] Christensen, L. W., Foxby, H.-B., Frankild, A.: Restricted homological dimensions and \hbox{Cohen}-Macaulayness. J. Algebra 251 (2002), 479-502. DOI 10.1006/jabr.2001.9115 | MR 1900297 | Zbl 1073.13501
[6] Green, E. L., Kirkman, E., Kuzmanovich, J.: Finitistic dimensions of finite-dimensional monomial algebras. J. Algebra 136 (1991), 37-50. DOI 10.1016/0021-8693(91)90062-D | MR 1085118 | Zbl 0727.16003
[7] Smal{ø, S. O.: Homological differences between finite and infinite dimensional representations of algebras. Infinite Length Modules. Proceedings of the Conference, Bielefeld, Germany, 1998 H. Krause et al. Trends Math. Birkhäuser, Basel (2000), 425-439. MR 1798916 | Zbl 0991.16002
[8] Wei, J.: Finitistic dimension and restricted flat dimension. J. Algebra 320 (2008), 116-127. DOI 10.1016/j.jalgebra.2008.03.017 | MR 2417981 | Zbl 1160.16003
[9] Xi, C.: On the finitistic dimension conjecture. II. Related to finite global dimension. Adv. Math. 201 (2006), 116-142. DOI 10.1016/j.aim.2005.02.002 | MR 2204752 | Zbl 1103.18011
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