We study the Betti cone of modules of a standard graded algebra over a field, and find a class of modules whose Betti diagrams span extremal rays of the Betti cone. We also identify a class of one-dimensional Cohen-Macaulay rings with certain Hilbert series, for which the Betti cone is spanned by these extremal rays. These results lead to an algorithm for the decomposition of Betti table of modules, into the extremal rays, over such rings, and also help to obtain bounds for the multiplicity of the given module.
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