Normal domains with monomial presentations

TL;DR

This paper characterizes when finitely generated monomial presentation algebras are integrally closed domains and computes their class groups, focusing on cases with up to two relations.

Contribution

It provides a purely relation-based characterization of integrally closed domains for monomial presentations with at most two relations and calculates their class groups.

Findings

Characterization of integrally closed domains in monomial presentations with ≤2 relations

Explicit calculation of class groups for these algebras

Conditions expressed solely in terms of defining relations

Abstract

Let A be a finitely generated commutative algebra over a field K with a presentation A=K < X_{1}, ..., X_{n} | R >, where R is a set of monomial relations in the generators X_{1}, ..., X_{n}. So A = K[S], the semigroup algebra of the monoid S=< X_{1}, ..., X_{n} | R >. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relations. Also the class group of such algebras A is calculated.