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

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 Nn.