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