The structure of the Boij-S\"oderberg posets

TL;DR

This paper investigates the structure of a specific family of posets related to Betti diagrams, proving they are bounded complete lattices with Cohen-Macaulay order complexes, advancing understanding in algebraic geometry.

Contribution

The paper proves that the Boij-S"oderberg posets are bounded complete lattices and their order complexes are vertex-decomposable, Cohen-Macaulay, and squarefree glicci.

Findings

Posets are bounded complete lattices

Order complexes are vertex-decomposable

Order complexes are Cohen-Macaulay and squarefree glicci

Abstract

Boij and S\"oderberg made a pair of conjectures, which were subsequently proven by Eisenbud and Schreyer and then extended by Boij and S\"oderberg, about the structure of Betti diagrams of Graded modules. In the theory, a particular family of posets, and their associated order complexes, play an integral role. We explore the structure of this family. In particular, we show the posets are bounded complete lattices and the order complexes are vertex-decomposable, hence Cohen-Macaulay and squarefree glicci.