Tight closure in graded rings

Abstract

. This paper facilitates the computation of tight closure by giving giving upper and lower bounds on the degrees of elements that need to be checked for inclusion in the tight closure of certain homogeneous ideals in a graded ring. Differential operators are introduced to the study of tight closure, and used to prove that the degree of any element in the tight closure of a homogeneous ideal (but not in the ideal itself) must exceed the minimal degree of the generators of the ideal. Briancon-Skoda-type theorems are used to give explicit bounds (in terms of the degrees of the generators of the ideal) such that all elements of at least this degree are in the tight closure of this homogeneous ideal. These ideas also yield a new test for a Cohen-Macaulay ring to be F-rational (and hence a rational singularity) in terms of its a-invariant alone. In its principal setting, tight closure is an operation performed on ideals in a commutative, Noetherian ring of prime characteristic. Th...