Adjoints of ideals in regular local rings (with an appendix by Steven Dale Cutkosky) (Q1897744)

Let \(I\) be an ideal in a regular local ring \(R\). The author associates with \(I\) an integrally closed ideal \(\widetilde I\). Several Briançon-Skoda type theorems are proved and used to improve previous results. For example, if the ideal \(I\) is generated by \(\ell\) elements, then \(\widetilde {I^{n + \ell}} \subseteq I^n\) for all \(n \geq 0\). Several conjectures are formulated, motivated and proved in some special cases. One of these conjectures is the vanishing conjecture (related to Grauert-Riemenschneider vanishing), which asserts that certain cohomology groups are zero. This conjecture is obtained in the appendix for rings essentially of finite type over a field of characteristic zero, as a consequence of a more general vanishing theorem. The paper also elaborates on the two-dimensional case.