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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
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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 Betti numbers, flat dimensions, local cohomology modules
MSC Classifications: 13D45, 13D05 show english descriptions Local cohomology [See also 14B15]
Homological dimension
13D45 - Local cohomology [See also 14B15]
13D05 - Homological dimension
 

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