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Let S be a polynomial ring in n variables over a field K of any characteristic. Given a strongly stable submodule M of a finitely generated graded free S-module F, we propose a method for constructing a standard Borel-fixed submoduleM ˜of F so that the extremal Betti numbers of M, values as well as positions, are preserved by passing from M toM ˜    . As a result, we obtain a numerical characterization of all possible extremal Betti numbers of any standard Borel-fixed submodule of a finitely generated graded free S-module F

A Constructive Method for Standard Borel Fixed Submodules with Given Extremal Betti Numbers