Gluing semigroups and strongly indispensable free resolutions

TL;DR

This paper investigates the strong indispensability of minimal free resolutions in semigroup rings, focusing on operations like gluing and extending to generate new examples with this property.

Contribution

It provides a naive criterion for the strong indispensability of glued semigroup rings and constructs numerous examples with this property, including infinite classes of complete intersection semigroups.

Findings

Identified conditions for strong indispensability in glued semigroup rings

Constructed infinite families of complete intersection semigroups with this property

Extended known classes of semigroups with strongly indispensable resolutions

Abstract

We study strong indispensability of minimal free resolutions of semigroup rings. We focus on two operations, gluing and extending, used in literature to produce more examples with a special property from the existing ones. We give a naive condition to determine whether gluing of two semigroup rings has a strongly indispensable minimal free resolution. As applications, we determine extensions of $3$-generated non-symmetric, $4$-generated symmetric and pseudo symmetric numerical semigroups as well as obtain infinitely many complete intersection semigroups of any embedding dimension, having strongly indispensable minimal free resolutions.