A constructive method for standard borel fixed submodules with given extremal Betti numbers

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 submodule $widetilde M$ of $F$ so that the extremal Betti numbers of $M$, values as well as positions, are preserved by passing from $M$ to $widetilde M$. 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 $F$.