Serre's Condition Rℓ for Affine Semigroup Rings

Abstract

In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition Rℓ 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 non-normal affine semigroup rings that satisfy R2.