Extremal Betti numbers of graded modules

TL;DR

This paper provides a complete numerical characterization of all extremal Betti numbers for graded submodules of finitely generated graded free modules over polynomial rings in characteristic zero.

Contribution

It introduces a comprehensive numerical framework to identify all possible extremal Betti numbers in this algebraic setting.

Findings

Characterization of extremal Betti numbers achieved

Applicable to all graded submodules of finitely generated free modules

Framework valid over polynomial rings in characteristic zero

Abstract

Let $S$ be a polynomial ring in $n$ variables over a field $K$ of characteristic $0$. A numerical characterization of all possible extremal Betti numbers of any graded submodule of a finitely generated graded free $S$-module is given.