On the Normality of a Class of Monomial Ideals via the Newton Polyhedron

摘要

Let I = x a1 1,..., x an n . R = K[ x1,..., xn] with a1,..., an positive integers and K a field, and let J be the integral closure of I. A criterion for the normality of J is developed. This criterion is used to show that J is normal if and only if the integral closure of the ideal x b1 1,..., x bn n,..., x br r . R[ xn+ 1,..., xr] is normal, where bi. {a1,..., an} for all i, this generalizes the work of Al-Ayyoub (Rocky Mt Math 39(1): 1-9, 2009). If l = lcm(a1,..., an) and the integral closure of x a1 1,..., x an n, x l n+ 1 . R[ xn+ 1] is not normal, then we show that the integral closure of x a1 1,..., x an n, x s n+ 1 is not normal for any s > l. Also, we give a shorter proof of a main result of Coughlin (Classes of Normal Monomial Ideals. Ph. D. thesis, 2004).