Serre's Condition R_l for Affine Semigroup Rings

TL;DR

This paper characterizes when affine semigroup rings satisfy Serre's R_l condition using face lattice structures and explores properties of Rees algebras of specific monomial ideals, including examples of nonnormal rings satisfying R_2.

Contribution

It provides a new characterization of Serre's R_l condition for affine semigroup rings based on face lattice analysis and examines Rees algebras of certain monomial ideals.

Findings

Characterization of R_l condition via face lattice of pos(S)

Rees algebras of specific monomial ideals often satisfy R_l

Examples of nonnormal rings satisfying R_2

Abstract

In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition R_l of Serre. Our characterization is in terms of the face lattice of the positive cone pos(S) of S. We start by reviewing some basic facts about the faces of pos(S) and consequences for the monomial primes of K[S]. After proving our characterization we turn our attention to the Rees algebras of a special class of monomial ideals in a polynomial ring over a field. In this special case, some of the characterizing criteria are always satisfied. We give examples of nonnormal affine semigroup rings that satisfy R_2.