Fundamental Results on s-Closures

This paper establishes the fundamental properties of the s-closures, a recently introduced family of closure operations on ideals of rings of positive characteristic. The behavior of the s-closure of homogeneous ideals in graded rings is studied, and criteria are given for when the s-closure of an ideal can be described exactly in terms of its tight closure and rational powers. Sufficient conditions are established for the weak s-closure to equal to the s-closure. A generalization of the Briancon-Skoda theorem is given which compares any two different s-closures applied to powers of the same ideal