The structure of the minimal free resolution of semigroup rings obtained by gluing

TL;DR

This paper provides an explicit construction for the minimal free resolution of semigroup rings obtained by gluing, enabling computation of algebraic invariants and extending results to higher dimensions.

Contribution

It introduces a method to construct minimal free resolutions of glued semigroup rings from those of the original semigroups, with applications to algebraic invariants.

Findings

Computed Betti numbers, regularity, and Hilbert series for glued semigroup rings.

Proved the differential graded algebra structure of the resolutions under certain conditions.

Extended the results to semigroups in N^n.

Abstract

We construct a minimal free resolution of the semigroup ring k[C] in terms of minimal resolutions of k[A] and k[B] when <C> is a numerical semigroup obtained by gluing two numerical semigroups <A> and <B>. Using our explicit construction, we compute the Betti numbers, graded Betti numbers, regularity and Hilbert series of k[C], and prove that the minimal free resolution of k[C] has a differential graded algebra structure provided the resolutions of k[A] and k[B] possess them. We discuss the consequences of our results in small embedding dimensions. Finally, we give an extension of our main result to semigroups in N^n