Laura Matusevich, Texas A&M University
2023 AWM Research Symposium
Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]

Affine semigroup rings are algebras (over a field) generated by finitely many Laurent monomials. Such rings are very amenable to combinatorial treatment, especially methods from polyhedral geometry and integer programming. This makes them attractive as examples for computations in commutative algebra, since they can exhibit a variety of behaviors that can be characterized in combinatorial terms. A key feature of affine semigroup rings is that they are finely graded, and as such, it makes sense to study their homogeneous ideals, namely the monomial ideals. It is more difficult to study monomial ideals in semigroup rings than it is to study monomial ideals in the polynomial ring. A first reason is that affine semigroup rings are not UFDs in general, and so it is challenging to do computations with generators. In this talk I will describe some known results on monomial ideals in affine semigroup rings, compare them with counterparts over the polynomial ring, and indicate some of the many open directions one could take.

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