Tight Closure in Graded Rings

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. Briann con-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 com-mutative, Noetherian ring of prime characteristic. This operation was introduced by Hochster and Huneke in HH1], and has had applications to several disparate but classical problems in commutative algebra such as the Syzygy problem, the local cohomological conjectures, and the Briann con-Skoda theorems. Tight closure appears to be giving information about the singularities of a local ring. For example , with mild hypotheses, the property that all ideals of a ring are tightly closed implies that ring is normal and Cohen-Macaulay HH1] and even pseudo-rational S1], which amounts to rational singularities in characteristic zero. Tight closure also sheds light on log terminal and log canonical singularities W] H]. However, a serious diiculty in this theory remains: how does one compute the tight closure of a given ideal in a given ring? This paper attacks the problem of computing the tight closure of homogeneous ideals in a graded ring. Because of the subtle information tight closure provides about both the ring and the ideal, an actual algorithm for computing tight closure seems much too much to hope for. However, it is of interest to at least narrow the search. In this paper, the problem is confronted from both ends. A general lower bound on the degrees of elements in I is proven (Theorems 2.2, 2.4): (with mild assumptions on R) any element in I ? I must have degree strictly larger than the smallest degree of any of the minimal …