Difference of Hilbert series of homogeneous monoid algebras and their normalizations

Abstract

Let Q be an affine monoid, \(\Bbbk [Q]\) the associated monoid \(\Bbbk \) -algebra, and \(\Bbbk [\overline{Q}]\) its normalization, where we let \(\Bbbk \) be a field. We discuss a difference of the Hilbert series of \(\Bbbk [Q]\) and \(\Bbbk [\overline{Q}]\) in the case where \(\Bbbk [Q]\) is homogeneous (i.e., standard graded). More precisely, we prove that if \(\Bbbk [Q]\) satisfies Serre’s condition \((S_2)\) , then the degree of the h-polynomial of \(\Bbbk [Q]\) is always greater than or equal to that of \(\Bbbk [\overline{Q}]\) . Moreover, we also show counterexamples of this statement if we drop the assumption \((S_2)\) .