Some results on normal homogeneous ideals

摘要

In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an N-graded ring A of the form A(greater than or equal tom):=circle plus(lgreater than or equal tom)A(l) and monomial ideals in a polynomial ring over a field. For ideals of the form A,,,, we generalize a recent result of Faridi. We prove that a monomial ideal in a polynomial ring in n indeterminates over a field is normal if and only if the first n - 1 positive powers of the ideal are integrally closed. We then specialize to the case of ideals of the form I(lambda):=, where J(lambda) = (x(1)(lambda1), . . . ,x(n)(lambdan)) subset of or equal to Kx(1), . . . ,x(n). To state our main result in this setting, we let l=lcm(lambda(1), . . . ,lambda(i), . . . lambda(n)), for 1 less than or equal to i less than or equal to n, and set lambda' = (lambda(1), . . . ,lambda(i-1), lambda(i) + l, lambda(i+1), . . . , lambda(n)). We prove that if I(lambda') is normal then I(lambda) is normal and that the converse holds with a small additional assumption. References: 10