Betti Numbers and Flat Dimensions of Local Cohomology Modules

摘要

Assume that R is a commutative Noetherian ring with non-zero identity, alpha 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 H-alpha(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 H-alpha(i) (X) in terms of the flat dimensions of the modules H-alpha(j) (X), j not equal i, and that of X.