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# Betti Numbers and Flat Dimensions of Local Cohomology Modules

Published:2015-06-02
Printed: Sep 2015
• Alireza Vahidi,
Department of Mathematics , Payame Noor University (PNU) , IRAN
 Format: LaTeX MathJax PDF

## Abstract

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 $\operatorname{H}_\mathfrak{a}^i(X)$. 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 $\operatorname{H}_\mathfrak{a}^i(X)$ in terms of the flat dimensions of the modules $\operatorname{H}_\mathfrak{a}^j(X)$, $j\not= i$, and that of $X$.
 Keywords: Betti numbers, flat dimensions, local cohomology modules
 MSC Classifications: 13D45 - Local cohomology [See also 14B15] 13D05 - Homological dimension

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