Infinite free resolutions over numerical semigroup algebras via specialization

TL;DR

This paper explores the structure of infinite free resolutions over numerical semigroup algebras, revealing how stratifications of semigroups influence homological properties and providing classifications in specific cases.

Contribution

It introduces a new combinatorial approach to understanding infinite free resolutions and their homological properties via stratifications of the Kunz cone.

Findings

Resolutions respect the Kunz cone stratification

Complete classification for the case m=4

Associated graded algebras do not respect the stratification

Abstract

Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in a polyhedral cone $C_m$, known as the Kunz cone. The faces of $C_m$ form a stratification of numerical semigroups that has been shown to respect a number of algebraic properties of $S$, including the combinatorial structure of the minimal free resolution of the defining toric ideal $I_S$. In this work, we prove that the structure of the infinite free resolution of the ground field $\Bbbk$ over the semigroup algebra $\Bbbk[S]$ also respects this stratification, yielding a new combinatorial approach to classifying homological properties like Golodness and rationality of the poincare series in this setting. Additionally, we give a complete classification of such resolutions in the special case $m = 4$, and demonstrate that the associated graded algebras do not generally respect the same stratification.