Normality of subrings generated by square free monomials

Abstract

Let R = k[ x 1, …, x n ] and R[ x] be a polynomial ring over a field k and let I be a normal ideal of R generated by square free monomials of the same degree. We prove that I + x( x 1, …, x n ) and I + ( xx 1) are both normal ideals of R[ x]. The ideals I t , and I t + xI t − 2 are shown to be normal, where I t , is the ideal of R generated by the square free monomials of degree t. Let P be the ideal of relations of the semigroup ring k[x ix j ¦ 1 ≤ i < j ≤ n] . We prove that the toric ideal P has a quadratic reduced Gröbner basis with respect to a lexicographical ordering, then we uncover some classes of Cohen-Macaulay rings and compute their Hubert series.