Geodesic
Finitistic dimension and restricted injective dimension
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 4, pp. 1023-1031
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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.
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.
DOI : 10.1007/s10587-015-0225-y
Classification : 18G10, 18G20
Keywords: finitistic dimension; restricted injective dimension; tilting module 
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Wu, Dejun. Finitistic dimension and restricted injective dimension. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 4, pp. 1023-1031. doi: 10.1007/s10587-015-0225-y

[1] Angeleri-Hügel, L., Trlifaj, J.: Tilting theory and the finitistic dimension conjectures. Trans. Am. Math. Soc. 354 (2002), 4345-4358. | DOI | MR | Zbl

[2] Asadollahi, J., Salarian, S.: Gorenstein injective dimension for complexes and Iwanaga-Gorenstein rings. Commun. Algebra 34 (2006), 3009-3022. | DOI | MR | Zbl

[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 | Zbl

[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 | Zbl

[5] Christensen, L. W., Foxby, H.-B., Frankild, A.: Restricted homological dimensions and \hbox{Cohen}-Macaulayness. J. Algebra 251 (2002), 479-502. | DOI | MR | Zbl

[6] Green, E. L., Kirkman, E., Kuzmanovich, J.: Finitistic dimensions of finite-dimensional monomial algebras. J. Algebra 136 (1991), 37-50. | DOI | MR | Zbl

[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 | Zbl

[8] Wei, J.: Finitistic dimension and restricted flat dimension. J. Algebra 320 (2008), 116-127. | DOI | MR | Zbl

[9] Xi, C.: On the finitistic dimension conjecture. II. Related to finite global dimension. Adv. Math. 201 (2006), 116-142. | DOI | MR | Zbl

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