Infinite free resolutions over numerical semigroup algebras via specialization

Each numerical semigroup S with smallest positive element m corresponds to an integer point in a polyhedral cone Cₘ, known as the Kunz cone. The faces of Cₘ 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 IS. In this work, we prove that the structure of the infinite free resolution of the ground field over the semigroup algebra 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.