F-rational rings and the integral closure of ideals
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The history of the Briançon-Skoda theorem and its ensuing avatars in commutative algebra has been well-documented in many papers. A theorem proved by Briançon and Skoda for convergent power series over the complex numbers and generalized to arbitrary regular local rings by Lipman and Sathaye states: Theorem 1.2 Let R be a regular local ring and let I be an ideal generated by ℓ elements. Then for all n ≥ ℓ, In ⊆ In−ℓ+1. This was partially extended to the class of pseudo-rational rings by Lipman and Teissier. However, they were unable to recover the full strength of (1.2). The present two authors, as well as Lipman, have pushed the original theorem further by introducing 'coefficients.' The methods used by the present authors have relied on the theory of tight closure. These improvements, however, have been valid only in regular rings, and the question of whether the statement of Theorem 1.2 remains valid in arbitrary pseudo-rational rings has remained open since 1981. Recent progress was made by Hyry and Villamayor, who proved (among other things) that if R is local Gorenstein and essentially of finite type over a field of characteristic 0, then In+ℓ−1 ⊆ In for an arbitrary ideal I with ℓ generators. In this paper we will use tight closure methods to prove (1.2) is valid for F-rational rings.