On decomposing Betti tables and $O$-sequences

TL;DR

This paper develops explicit formulas for decomposing Betti tables of modules and ideals, linking Boij-S"oderberg decompositions with $O$-sequences, and characterizes the cone of $O$-sequences.

Contribution

It introduces explicit extension formulas for Betti table decompositions and describes the convex cone of $O$-sequences, advancing understanding of their structure.

Findings

Explicit formulas for Betti table decompositions

Connection between $O$-sequences and Betti table coefficients

Description of the convex cone of $O$-sequences

Abstract

The Boij-S\"oderberg characterization decomposes a Betti table into a unique positive integral linear combination of pure diagrams. Given a module with a pure resolution, we describe explicit formulae for computing the decomposition of the Betti table of the module given the decomposition of the truncation of the Betti table, and vice versa. Nagel and Sturgeon described the decomposition of Betti tables of ideals with $d$-linear resolutions; indeed, the coefficients are precisely finite $O$-sequences. Using the extension formulae, we provide an explicit description of the coefficients of the decomposition of the Betti table of the quotient ring of such an ideal. Following from this, we describe the closed convex simplicial cone of $O$-sequences.