The Semigroup of Betti Diagrams

TL;DR

This paper explores the algebraic structure of Betti diagrams, proving that the set of Betti diagrams forms a finitely generated semigroup, thus advancing understanding of their combinatorial properties.

Contribution

It establishes that the semigroup of Betti diagrams is finitely generated and addresses key foundational questions about its structure.

Findings

The semigroup of Betti diagrams is finitely generated.

Several fundamental questions about the semigroup are answered.

Provides new insights into the algebraic and combinatorial structure of Betti diagrams.

Abstract

The recent proof of the Boij-Soederberg conjectures reveals new structure about Betti diagrams of modules, giving a complete description of the cone of Betti diagrams. We begin to expand on this new structure by investigating the semigroup of Betti diagrams. We prove that this semigroup is finitely generated, and we answer several other fundamental questions about this semigroup.