Betti Numbers and Flat Dimensions of Local Cohomology Modules

Vahidi, Alireza

Canadian mathematical bulletin, Tome 58 (2015) no. 3, pp. 664-672

Résumé

Assume that $R$ is a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ is an ideal of $R$ , and $X$ is an $R$ -module. In this paper, we first study the finiteness of Betti numbers of local cohomology modules $\text{H}_{\mathfrak{a}}^{i}\left( X \right)$ . Then we give some inequalities between the Betti numbers of $X$ and those of its local cohomology modules. Finally, we present many upper bounds for the flat dimension of $X$ in terms of the flat dimensions of its local cohomology modules and an upper bound for the flat dimension of $\text{H}_{\mathfrak{a}}^{i}\left( X \right)$ in terms of the flat dimensions of the modules $\text{H}_{\mathfrak{a}}^{j}\left( X \right),j\ne i$ , and that of $X$ .

Détail

DOI : 10.4153/CMB-2015-042-7

Mots-clés : 13D45, 13D05, Betti numbers, flat dimensions, local cohomology modules

Bibliographie

[1] [1] Aghapournahr, M., Taherizadeh, A. J., and Vahidi, A., Extension functors of local cohomology modules. Bull. Iran. Math. Soc. 37(2011), no. 3,117–134. Google Scholar

[2] [2] Brodmann, M. P. and Sharp, R. Y., Local cohomology: an algebraic introduction with geometric applications. Cambridge Studies in Advanced Mathematics, 60, Cambridge University Press, Cambridge, 1998. Google Scholar

[3] [3] Bruns, W. and Herzog, J., Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993. Google Scholar

[4] [4] Delfino, D. and Marley, T., Cofinite modules and local cohomology. J. Pure Appl. Algebra 121(1997), no. 1, 45–52. http://dx.doi.Org/10.101 6/S0022-4049(96)00044-8 Google Scholar

[5] [5] Dibaei, M. T. and Vahidi, A., Artinian and non-Artinian local cohomology modules. Canad. Math. Bull. 54(2011), no. 4, 619–629. Google Scholar | DOI

[6] [6] Dibaei, M. T. and Vahidi, A., Torsion functors of local cohomology modules. Algebr. Represent. Theory 14(2011), no. 1, 79–85. http://dx.doi.Org/10.1007/s10468-009-91 77-y Google Scholar

[7] [7] Dibaei, M. T. and Yassemi, S., Bass numbers of local cohomology modules with respect to an ideal. Algebr. Represent. Theory H(2008), no. 3, 299–306. http://dx.doi.Org/10.1007/s10468-007-9072-3 Google Scholar

[8] [8] Hartshorne, R., Cohomological dimension of algebraic varieties. Ann. of Math. 88(1968), 403–450. http://dx.doi.Org/10.2307/1 970720 Google Scholar

[9] [9] Hassanzadeh, S. H. and Vahidi, A. On vanishing and cofiniteness of generalized local cohomologymodules. Commun. Algebra 37(2009), no. 7, 2290–2299. Google Scholar | DOI

[10] [10] Kawasaki, K., On the finiteness of Bass numbers of local cohomology modules. Proc. Amer. Math. Soc. 124(1996), no. 11, 3275–3279. Google Scholar | DOI

[11] [11] Rotman, J., An introduction to homological algebra. Pure and Applied Mathematics, 85, Academic Press, New York-London, 1979. Google Scholar