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Betti Numbers and Flat Dimensions of Local Cohomology Modules

Published online by Cambridge University Press:  20 November 2018

Alireza Vahidi
Alireza Vahidi*
Affiliation:
Department of Mathematics, Payame Noor University (PNU), IRAN e-mail: vahidi.ar@pnu.ac.ir
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Abstract

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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 $X$.

Keywords

13D4513D05Betti numbersflat dimensionslocal cohomology modules
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 3 , 01 September 2015 , pp. 664 - 672
Copyright
Copyright © Canadian Mathematical Society 2015

References

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