Document Type : Original Manuscript


Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran.



Let $(R, \mathfrak{m})$ be a Noetherian local ring and $M$, $N$ be two finitely generated $R$-modules. In this paper it is shown that $R$ is a Cohen-Macaulay ring if and only if $R$ admits a non-zero Artinian $R$-module $A$ of finite projective dimension; in addition, for all such Artinian $R$-modules $A$, it is shown that $\mathrm{pd}_R\, A=\dim R$. Furthermore, as an application of these results it is shown that
$$\pdd H^i_{{\frak p}R_{\frak p}}(M_{\frak p}, N_{\frak p})\leq \pd H^{i+\dim R/{\frak p}}_{\frak m}(M,N)$$
for each ${\frak p}\in \mathrm{Spec} R$ and each integer $i\geq 0$. This result answers affirmatively a question raised by the present authors in [13].


1. J. Asadollahi, K. Khashyarmanesh, and S. Salarian, On the finiteness properties of the generalized local cohomology modules, Commun. Algebra, 30 (2002), 859–867.

2. M. Asgharzadeh, K. Divaani-Aazar, and M. Tousi, The finiteness dimension of local cohomology modules and its dual notion, J. Pure. Appl. Algebra, 320 (2008), 1275–1287.

3. K. Bahmanpour, A note on homological dimensions of Artinian local cohomology modules, Canad. Math. Bull., 56 (2013), 491–499.

4. M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J., 21 (1980), 173–181.

5. K. Borna, P. Sahandi, and S. Yassemi, Cofiniteness of generalized local cohomology modules, Bull. Aust. Math. Soc., 83 (2011), 382–388.

6. M. P. Brodmann, and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge University Press, Cambridge, 1998.

7. L. Chu, Cofiniteness and finiteness of generalized local cohomology modules,
Bull. Aust. Math. Soc., 80 (2009), 244–250.

8. K. Divaani-Aazar, and A. Hajikarimi, Cofiniteness of Generalized local cohomology modules for one-dimensional ideals, Canad. Math. Bull., 55 (2012), 81–87.

9. K. Divaani-Aazar, and R. Sazeedeh, Cofiniteness of generalized local cohomology modules, Colloq. Math., 99 (2004), 283–290.

10. K. Divaani-Aazar, R. Sazeedeh, and M. Tousi, On vanishing of generalized local cohomology modules, Algebra Colloq., 12 (2005), 213–218.

11. C.U. Jensen, On the vanishing of lim (i), J. Algebra, 15, (1970), 151–166.

12. J. Herzog, Komplexe, Auflösungen und Dualität in der lokalen Algebra, Habilitationsschrift,
Universität Regensburg, 1974.

13. Y. Irani, K. Bahmanpour, and G. Ghasemi, Some homological propertis of generalized local cohomology modules, J. Algebra Appl., 18 (2019), Article ID: 1950238, (10 pages).

14. H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, UK, 1986.

15. L. Gruson and M. Raynaud, Critères de platitude et de projectivité Techniques de ′′platification′′ d′un module., Invent. Math., 13 (1971), 1–89.

16. C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Publ. Math. I.H.E.S. (Paris) 42 (1973), 47–119.


17. N. Suzuki, On the generalized local cohomology and its duality, J. Math. Kyoto Univ., 18 (1978), 71–78.

18. S. Yassemi, Generalized section functors, J. Pure. Appl. Algebra, 95 (1994), 103–119.

19. S. Yassemi, L. Khatami, and T. Sharif, Associated primes of generalized local cohomology modules, Commun. Algebra, 30 (2002), 327–330.


20. N. Zamani, On graded generalized local cohomology, Arch. Math., 86 (2006), 321–330.